^a^ainstitutetext: Department of Physics, Key Laboratory of Low Dimensional Quantum Structures and Quantum Control of Ministry of Education, and Synergetic Innovation Center for Quantum Effects and Applications, Hunan Normal University, 36 Lushan Road, Changsha, Hunan 410081, People’s Republic of China^b^binstitutetext: School of Physics and Astronomy, China West Normal University, 1 Shida Road, Nanchong, Sichuan 637002, People’s Republic of China

Thermodynamic topological classes of the rotating, accelerating black holes

Wentao Liu a Li Zhang b,1 Di Wu¹¹1Corresponding author. a,1 and Jieci Wang [email protected], [email protected], [email protected], [email protected]

Abstract

In this paper, we extend our previous work [D. Wu, Phys. Rev. D 108, 084041 (2023)] to more general cases by including a rotation parameter. We investigate the topological numbers for the rotating, accelerating neutral black hole and its AdS extension, as well as the rotating, accelerating charged black hole and its AdS extension. We find that the topological number of an asymptotically flat accelerating black hole consistently differs by one from that of its non-accelerating counterpart. Furthermore, we show that for an asymptotically AdS accelerating black hole, the topological number is reduced by one compared to its non-accelerating AdS counterpart. In addition, we demonstrate that within the framework of general relativity, the acceleration parameter and the negative cosmological constant each independently add one to the topological number. However, when both factors are present, their effects neutralize each other, resulting in no overall change to the topological number.

Keywords:

Black Holes, Spacetime Singularities, AdS-CFT Correspondence

1 Introduction

The study of the properties of black hole spacetimes and gravitational fields fundamentally relies on exact solutions to the gravitational field equations. Among the well-known family of exact black hole solutions to the four-dimensional Einstein field equations in General Relativity (GR), the most general Petrov type-D electrovacuum black hole solution is the Plebanski-Demianski metric AP98-98 . The simplest case of this metric, non-rotating, uncharged, and without NUT charge or cosmological constant, is referred to as the C-metric PRD2-1359 ; IJMPD15-335 . This class of solutions is also known as the accelerating black holes, which were interpreted by Kinnersley and Walker PRD2-1359 as the gravitational field produced by a point mass undergoing uniform acceleration, i.e., a pair of black holes accelerating away from each other in opposite directions GRG15-535 . In recent years, a wide range of studies has focused on the different features of accelerating black holes, including their global causal structure CQG23-6745 , quasinormal modes PRD109-084049 ; JHEP1022047 ; JHEP0224140 ; JHEP0224189 , quantum thermal properties PLA209-6 , holographic heat engines EPJC78-645 ; JHEP0219144 , weak/strong cosmic censorship conjecture PLB849-138433 ; SCPMA66-280412 , black hole shadows PRD103-025005 , and holographic complexity PLB823-136731 ; PLB838-137691 , etc JHEP0111114 ; JHEP1214057 ; JHEP0916082 ; PRL126-111601 ; JHEP0122102 ; PRL130-091603 ; JHEP0522063 ; JHEP1022074 ; JHEP0923122 ; JHEP1123073 ; JHEP0324050 ; JHEP0324079 ; 2404.04691 . Especially, in Refs. PRL117-131303 ; JHEP0517116 ; PRD98-104038 ; JHEP0419096 ; PLB796-191 ; CQG38-145031 ; CQG38-195024 ; PRD104-086005 ; 2306.16187 ; 2407.21329 , the thermodynamics of four-dimensional accelerating (charged and rotating) AdS black holes was thoroughly studied, leading to extensions of the first law of thermodynamics PRD7-2333 ; PRD13-191 , the Bekenstein-Smarr mass formula PRL30-71 , and the Christodoulou-Ruffini-like squared-mass formula PRL25-1596 ; PRD4-3552 ; PRD101-024057 ; PRD102-044007 ; PRD103-044014 ; JHEP1121031 to such spacetimes.²²2Recently, these formulas were perfectly extended to the Lorentzian Taub-NUT spacetimes PRD100-101501 ; PRD105-124013 ; PRD108-064034 ; PRD108-064035 ; PLB846-138227 .

The development of the above mass formulas represents just one facet of black hole thermodynamics. Quite recently, topology has emerged as a powerful mathematical tool, attracting substantial interest for its role in analyzing the thermodynamic phase transitions of black holes PRD105-104003 ; PRD105-104053 ; PLB835-137591 ; PRD107-046013 ; PRD107-106009 ; JHEP0623115 ; 2305.05595 ; 2305.05916 ; 2305.15674 ; 2305.15910 ; 2306.16117 ; PRD106-064059 ; PRD107-044026 ; PRD107-064015 ; 2212.04341 ; 2302.06201 ; 2304.14988 ; 2309.00224 ; 2312.12784 ; 2402.18791 ; 2403.14730 ; 2404.02526 ; 2407.09122 ; 2407.20016 ; 2408.03090 ; 2408.03126 ; NPB1006-116653 ; 2408.05870 .³³3Topology has also proven effective in the study of the properties of light rings PRL119-251102 ; PRL124-181101 ; PRD102-064039 ; PRD103-104031 ; PRD104-044019 ; PRD105-024049 ; PRD105-064070 ; PRD108-104041 ; PRD109-064050 ; 2408.05569 and timelike circular orbits PRD107-064006 ; JCAP0723049 ; 2406.13270 . In Ref. PRL129-191101 , Wei et al. proposed a novel and systematic classification scheme for black holes by considering them as thermodynamic topological defects, categorized according to their respective topological numbers in a pioneering manner. A brief account of this method is presented below.

Hawking temperature $T$ and Bekenstein-Hawking entropy $S$ are two very important quantities in black hole thermodynamics. In general, the standard Holmholtz free energy $F$ can be expressed as $F=M-TS$ PRD15-2752 ; PRD33-2092 ; PRD105-084030 ; PRD106-106015 . Inspired by this, we introduce the generalized off-shell Helmholtz free energy, which can be expressed in the following form PRL129-191101 :

\mathcal{F}=M-\frac{S}{\tau}\,,

(1)

where the extra variable $\tau$ can be interpreted as the inverse temperature of the cavity surrounding the black hole. The generalized Helmholtz free energy only exhibits on-shell characteristics and returns to the standard Helmholtz free energy of the black hole when $\tau=T^{-1}$ . Now, using Duan’s $\phi$ -mapping theory SS9-1072 ; NPB514-705 ; PRD61-045004 , one can construct a two-dimensional vector field $\phi=(\phi^{r},\phi^{\Theta})$ from the generalized Helmholtz free energy $\mathcal{F}$ , where

\displaystyle\phi=\Big{(}\frac{\partial\mathcal{F}}{\partial r_{h}}\,,~{}-\cot\Theta\csc\Theta\Big{)}\,,

(2)

in which $r_{h}$ is the radius of the black hole’s event horizon, and $\Theta$ is an additional factor, with $\Theta\in[0,+\infty]$ . It is important to note that the component $\phi^{\Theta}$ becomes infinite at $\Theta=0$ and $\Theta=\pi$ , indicating that the vector points outward in both cases. Therefore, the unit vector is formulated as $n=(n^{r},n^{\Theta})$ with $n^{r}=\phi^{r_{h}}/||\phi||$ and $n^{\Theta}=\phi^{\Theta}/||\phi||$ . The topological current $j^{\mu}$ satisfies the continuity equation $\partial_{\mu}j^{\mu}=0$ , indicating that it is conserved. This current only has a nonzero contribution at points where the vector field $\phi^{a}(x_{i})$ equals zero, ensuring that the topological charge is well-defined. For a given parameter region $\rho$ , the topological number is expressed as

W=\int_{\rho}j^{0}d^{2}x=\sum_{i=1}^{N}\beta_{i}\eta_{i}=\sum_{i=1}^{N}w_{i}\,,

(3)

where $w_{i}$ represents the winding number at the $i$ th point of $\phi$ , $j^{0}$ denotes the density of the topological current, $\beta_{i}$ is the positive Hopf index, and the Brouwer degree $\eta_{i}=\mathrm{sign}(J^{0}({\phi}/{x})_{z_{i}})=\pm 1$ , respectively. Note that the local winding number $w_{i}$ is regarded as a crucial tool for determining local thermodynamic stability. A positive $w_{i}$ suggests that the black holes are thermodynamically stable, while a negative $w_{i}$ indicates instability. The global topological number $W$ represents the difference between the counts of thermodynamically stable and unstable black holes at a specific temperature.

The elegance and general applicability of this method led to its widespread adoption, facilitating the exploration of topological numbers across different black hole models PRD107-064023 ; JHEP0123102 ; PRD107-024024 ; PRD107-084002 ; PRD107-084053 ; 2303.06814 ; 2303.13105 ; 2304.02889 ; 2306.13286 ; 2304.05695 ; 2306.05692 ; 2306.11212 ; EPJC83-365 ; 2306.02324 ; 2307.12873 ; 2309.14069 ; AP458-169486 ; 2310.09602 ; 2310.09907 ; 2310.15182 ; 2311.04050 ; 2311.11606 ; 2312.04325 ; 2312.06324 ; 2312.13577 ; 2312.12814 ; PS99-025003 ; 2401.16756 ; 2402.00106 ; PLB856-138919 ; AP463-169617 ; PDU44-101437 ; 2403.14167 ; PDU46-101617 ; 2405.02328 ; 2405.07525 ; 2405.20022 ; 2406.08793 ; AC48-100853 ; 2407.05325 ; 2408.08325 ; 2409.04997 ; EPJP139-806 . However, all previous research has primarily focused on non-accelerating cases, overlooking the topological numbers of accelerating black holes. Very recently, we have extended the topological approach to include non-rotating, accelerating black holes, considering both neutral and charged cases, as well as their AdS extensions PRD108-084041 . We have reached some intriguing conclusions: I) the topological number of an asymptotically flat, non-rotating, accelerating black hole always differs by one from that of its corresponding asymptotically flat, non-rotating, non-accelerating black hole; and II) the topological number of an asymptotically AdS, non-rotating, accelerating black hole consistently differs by minus one from that of its corresponding asymptotically AdS, non-rotating, non-accelerating black hole. Therefore, it makes sense to extend this work to more general rotating, accelerating black hole cases to investigate how the acceleration parameter affects the topological numbers of rotating black holes, and to verify whether these conclusions can be generalized to rotating, accelerating black holes. This also serves as the motivation for the current study.

In this paper, we explore, within the framework of the topological method for black hole thermodynamics, the topological numbers of the rotating, accelerating neutral black hole and its AdS counterpart, as well as the rotating, accelerating charged black hole and its AdS extension, via the generalized off-shell Helmholtz free energy. We found that, for the thermodynamic topological numbers of black hole solutions within the framework of general relativity, while both the acceleration parameter and the negative cosmological constant independently increase the topological number by one, their effects counterbalance each other when both are present in a black hole solution, which leads us to conjure that it might also hold true for the accelerating AdS black holes in alternative theories of gravity.

The remaining part of this paper is organized as follows. In section 2, we briefly introduce the rotating, accelerating charged AdS black hole. Section 3 begins with an exploration of the topological number of the rotating, accelerating black hole, focusing on the Kerr C-metric solution, and then extends this analysis to the Kerr-AdS C-metric solution, where a negative cosmological constant is present. In section 4, we examine the topological number of the rotating, accelerating charged black hole through the Kerr-Newman C-metric (KN C-metric) solution and subsequently extend this investigation to the KN-AdS C-metric case. Finally, section 5 presents our conclusions and future outlooks.

2 A brief introduction of rotating charged AdS accelerating black hole

Refer to caption — Figure 1: The structure of the family of black hole solutions represented by metric (4). This family has five parameters $m$ , $a$ , $q$ , $l$ , $A$ .

We start by introducing the generalized AdS C-metric solution, derived from the Plebanski-Demianski metric AP98-98 in Boyer-Lindquist type coordinates IJMPD15-335 , which includes rotation, electric charge, cosmological constant $\Lambda=-3/l^{2}$ , and the corresponding Abelian gauge potential, expressed as JHEP0419096

$\displaystyle ds^{2}$	$\displaystyle=$	$\displaystyle\frac{1}{H^{2}}\Bigg{\{}-\frac{f(r)}{\Sigma}\left(\frac{dt}{\alpha}-a\sin^{2}\theta\frac{d\varphi}{K}\right)^{2}+\frac{\Sigma}{f(r)}dr^{2}+\frac{\Sigma}{h(\theta)}d\theta^{2}$	(4)
		$\displaystyle+\frac{h(\theta)\sin^{2}\theta}{\Sigma}\left[\frac{adt}{\alpha}-(r^{2}+a^{2})\frac{d\varphi}{K}\right]^{2}\Bigg{\}}\,,$	(4)
$\displaystyle F$	$\displaystyle=$	$\displaystyle dB\,,\qquad B=-\frac{qr}{\Sigma}\left(\frac{dt}{\alpha}-a\sin^{2}\theta\frac{d\varphi}{K}\right)+\frac{qr}{(r^{2}+a^{2})\alpha}dt\,,$	(5)

where $\alpha$ is a rescaled factor, and the metric functions are

	$\displaystyle f(r)=(1-A^{2}r^{2})(r^{2}-2mr+a^{2}+q^{2})+\frac{r^{4}+a^{2}r^{2}}{l^{2}}\,,$
	$\displaystyle h(\theta)=1+2mA\cos\theta+\left[A^{2}(a^{2}+q^{2})+\frac{a^{2}}{l^{2}}\right]\cos^{2}\theta\,,$		(6)
	$\displaystyle\Sigma=r^{2}+a^{2}\cos^{2}\theta\,,\qquad H=1+Ar\cos\theta\,.$

in which $K$ is the conical deficit of the spacetime, $m$ , $a$ , $q$ , $A$ , $l$ are the mass, rotation, electric charge, acceleration parameters of the black hole and the AdS radius, respectively. The event horizon is located at the largest root of equation: $f(r_{h})=0$ . For the special case in which $a$ , $q$ , $A$ vanish and $l\to\infty$ , the general structure of this family of black hole solutions is given in figure 1. The $K$ factor is typically integrated into the azimuthal coordinate, resulting in an arbitrary periodicity, typically determined by a regularity condition at one of the poles. However, here the $K$ factor is explicitly included to ensure that the periodicity of $\varphi$ remains fixed at $2\pi$ . The conical deficit at each axis is then determined by analyzing the behavior of the $\theta$ - $\varphi$ portion of the metric near the south pole $\theta=\theta_{+}=0$ and the north pole $\theta=\theta_{-}=\pi$ , respectively JHEP0419096 :

ds_{\theta,\varphi}^{2}\propto d\theta^{2}+h^{2}(\theta)\sin^{2}\theta\frac{d\varphi^{2}}{K^{2}}\sim d\vartheta^{2}+(\Xi\pm 2mA)^{2}\vartheta^{2}d\varphi^{2}\,,

(7)

where $\vartheta_{\pm}=\pm(\theta-\theta_{\pm})$ provides a local radial coordinate near each axis, and

\Xi=1-\frac{a^{2}}{l^{2}}+A^{2}(q^{2}+a^{2})\,.

(8)

The above geometry indicates that there exists a conical singularity at the symmetry axis, with the tensions of the cosmic strings being

\mu_{\pm}=\frac{1}{4}\left(1-\frac{\Xi\pm 2mA}{K}\right)\,.

(9)

Thus, it is easy to see that the acceleration arises from the difference in conical deficits between the north and south poles:

\mu_{-}-\mu_{+}=\frac{mA}{K}\,,

(10)

where $K$ represents an overall deficit in the spacetime:

\bar{\mu}=\frac{1}{2}(\mu_{+}+\mu_{-})=\frac{1}{4}-\frac{\Xi}{4K}\,.

(11)

3 Topological classes of rotating neutral accelerating black hole

In this section, we will discuss the topological number of the four-dimensional rotating, accelerating neutral black hole by considering the Kerr C-metric black hole, and then extend the analysis to the case of the Kerr-AdS C-metric black hole with a nonzero negative cosmological constant.

3.1 Kerr C-metric black hole

The metric of the Kerr C-metric black hole is still given by Eq. (4), and now the metric functions $f(r)$ and $h(\theta)$ reduce to $f(r)=(1-A^{2}r^{2})(r^{2}-2mr+a^{2})$ and $h(\theta)=1+2mA\cos\theta+A^{2}a^{2}\cos^{2}\theta$ , respectively. The thermodynamic quantities are JHEP0419096

	$\displaystyle M=\frac{m(1-A^{2}a^{2})}{\alpha K(1+A^{2}a^{2})}\,,~{}\qquad T=\frac{\partial_{r_{h}}f(r_{h})}{4\pi\alpha(r_{h}^{2}+a^{2})}\,,~{}\qquad S=\frac{\pi(r_{h}^{2}+a^{2})}{K(1-A^{2}r_{h}^{2})}\,,$
	$\displaystyle\alpha=\frac{\sqrt{1-A^{2}a^{2}}}{\sqrt{1+A^{2}a^{2}}}\,,~{}\qquad J=\frac{ma}{K^{2}}\,,~{}\qquad\mu_{\pm}=\frac{1}{4}\left[1-\frac{1\pm 2mA+A^{2}a^{2}}{K}\right]\,,$		(12)
	$\displaystyle\Omega=\frac{aK}{\alpha(r_{h}^{2}+a^{2})}-\frac{A^{2}aK}{\alpha(1+A^{2}a^{2})}\,,\quad\lambda_{\pm}=\frac{r_{h}}{\alpha(1\pm Ar_{h})}-\frac{MK}{1+A^{2}a^{2}}\mp\frac{Aa^{2}}{\alpha(1+A^{2}a^{2})}\,.\quad$

It is a simple matter to check that the above thermodynamic quantities simultaneously satisfy the first law and the Bekenstein-Smarr mass formula

	$\displaystyle dM$	$\displaystyle=$	$\displaystyle TdS+\Omega dJ-\lambda_{+}d\mu_{+}-\lambda_{-}d\mu_{-}\,,$		(13)
	$\displaystyle M$	$\displaystyle=$	$\displaystyle 2TS+2\Omega\,J\,.$		(14)

The Gibbs free energy for this black hole can be determined through the computation of the Euclidean action as outlined below

\mathcal{I}_{E}=\frac{1}{16\pi}\int_{M}d^{4}x\sqrt{g}R+\frac{1}{8\pi}\int_{\partial{}M}d^{3}x\sqrt{h}(\mathcal{K}-\mathcal{K}_{0})\,.

(15)

Here, $h$ represents the determinant of the induced metric $h_{ij}$ , $\mathcal{K}$ denotes the extrinsic curvature of the boundary, and $\mathcal{K}_{0}$ corresponds to the extrinsic curvature of the massless C-metric solution, used as the reference background. The evaluation of the Euclidean action integral leads to the following expression for the Gibbs free energy

G=\frac{\mathcal{I}_{E}}{\beta}=\frac{m(1-A^{2}a^{2})}{2\alpha K(1+A^{2}a^{2})}=\frac{M}{2}=M-TS-\Omega\,J\,,

(16)

where $\beta=1/T$ being the interval of the time coordinate. Moreover, the final equality in Eq. (16) holds by applying the results from Eq. (3.1). Consequently, the conical singularity, characterized by the $(\lambda_{\pm}-\mu_{\pm})$ -pairs, does not influence the Gibbs free energy calculation for the Kerr C-metric black hole.

In the following, we will derive the topological number of the Kerr C-metric black hole. We note that the Helmholtz free energy is simply given by

F=G+\Omega\,J=M-TS\,.

(17)

Replacing $T$ with $1/\tau$ in Eq. (17) and utilizing $m=(r_{h}^{2}+a^{2})/(2r_{h})$ , then the generalized off-shell Helmholtz free energy is

\mathcal{F}=\frac{(r_{h}^{2}+a^{2})(1-A^{2}a^{2})}{2\alpha Kr_{h}(1+A^{2}a^{2})}-\frac{\pi(r_{h}^{2}+a^{2})}{K\tau(1-A^{2}r_{h}^{2})}\,,

(18)

Using the definition of Eq. (2), the components of the vector $\phi$ can be easily computed as follows:

	$\displaystyle\phi^{r_{h}}$	$\displaystyle=$	$\displaystyle\frac{(r_{h}^{2}-a^{2})(1-A^{2}a^{2})}{2\alpha Kr_{h}^{2}(1+A^{2}a^{2})}-\frac{2\pi r_{h}(1+A^{2}a^{2})}{K\tau(A^{2}r_{h}^{2}-1)^{2}}\,,$		(19)
	$\displaystyle\phi^{\Theta}$	$\displaystyle=$	$\displaystyle-\cot\Theta\csc\Theta\,.$		(20)

By solving the equation: $\phi^{r_{h}}=0$ , one can obtain the zero point of the vector field $\phi^{r_{h}}$ as

\tau=\frac{4\pi\alpha r_{h}^{3}(1+A^{2}a^{2})^{2}}{(r_{h}^{2}-a^{2})(1-A^{2}a^{2})(A^{2}r_{h}^{2}-1)^{2}}\,.

(21)

It is important to note that Eq. (21) consistently reduces to the result obtained in the case of the four-dimensional Kerr black hole PRD107-024024 when the acceleration parameter $A$ vanishes.

Assuming $Ar_{0}=0.5$ , $a/r_{0}=1$ for the Kerr C-metric black hole, we plot Figs. 2 and 3 to illustrate key aspects of the system. These figures display the zero points of the component $\phi^{r_{h}}$ and the behavior of the unit vector field $n$ on a segment of the $\Theta-r_{h}$ plane, with $\tau=20r_{0}$ , where $r_{0}$ is an arbitrary length scale defined by the size of the cavity enclosing the rotating, accelerating black hole.

From Fig. 2, it is apparent that for any given value of $\tau$ , there is only one thermodynamically stable Kerr C-metric black hole. This sharply contrasts with the four-dimensional Kerr black hole, which exhibits two distinct thermodynamic behaviors: one black hole branch that is stable and another that is unstable PRD107-024024 . This distinction underscores the significant influence of the acceleration parameter on the thermodynamical numbers of the rotating neutral black holes.

In Fig. 3, the zero point is found at $(r_{h}/r_{0},\Theta)=(3.44,\pi/2)$ . As a result, the winding number $w_{i}$ for the blue contour $C$ is identified as $w_{1}=1$ , which differ from those associated with the four-dimensional Kerr black hole PRD107-024024 . Regarding the global topological characteristics, the topological number $W=1$ for the Kerr C-metric black hole can be directly seen in Fig. 3, distinguishing it from the topological number of the Kerr black hole ( $W=0$ ). Therefore, it can be inferred that the Kerr C-metric black hole and the Kerr black hole differ not only in geometric topology but also belong to distinct categories in thermodynamic topology. Consequently, it would be compelling to investigate the topological characteristics of black holes with unusual geometries, such as the type-D NUT C-metric black hole 2409.06733 , though this first requires establishing its consistent thermodynamics. In addition, combining the intriguing conclusion presented in Ref. PRD108-084041 , which states that the difference in topological numbers between an asymptotically flat, static, accelerating black hole and its corresponding non-accelerating counterpart is always unity, we suggest a more general conjecture within the framework of general relativity. Specifically, we propose that the difference in topological numbers between any asymptotically flat accelerating black hole and its corresponding non-accelerating black hole is consistently unity. This inference will be further substantiated in section 4.1, where we compute the topological number of the KN C-metric black hole and compare it with that of the KN black hole, thereby demonstrating the validity and applicability of this conjecture.

3.2 Kerr-AdS C-metric black hole

In this subsection, we will extend the above discussions to the cases of the rotating, accelerating neutral AdS black hole by considering the Kerr-AdS C-metric black hole, whose metric is still given by Eq. (4), but now

	$\displaystyle f(r)$	$\displaystyle=(1-A^{2}r^{2})(r^{2}-2mr+a^{2})+\frac{r^{4}+a^{2}r^{2}}{l^{2}}\,,$		(22)
	$\displaystyle h(\theta)$	$\displaystyle=1+2mA\cos\theta+\left[A^{2}a^{2}+\frac{a^{2}}{l^{2}}\right]\cos^{2}\theta\,,$		(22)

where the AdS radius $l$ is associated with the thermodynamic pressure $P=3/(8\pi l^{2})$ of the four-dimensional AdS black holes CPL23-1096 ; CQG26-195011 ; PRD84-024037 .

The thermodynamic quantities are JHEP0419096

		$\displaystyle M=\frac{m(\Xi+a^{2}/l^{2})(1-A^{2}l^{2}\Xi)}{\Xi\alpha K(1+A^{2}a^{2})}\,,\quad T=\frac{\partial_{r_{h}}f(r_{h})}{4\pi\alpha(r_{h}^{2}+a^{2})}\,,~{}\quad S=\frac{\pi(r_{h}^{2}+a^{2})}{K(1-A^{2}r_{h}^{2})}\,,$		(23)
		$\displaystyle J=\frac{ma}{K^{2}}\,,\quad\Omega=\frac{aK}{\alpha(r_{h}^{2}+a^{2})}-\frac{aK(1-A^{2}l^{2}\Xi)}{l^{2}\Xi\alpha(1+A^{2}a^{2})}\,,\quad\alpha=\frac{\sqrt{(\Xi+a^{2}/l^{2})(1-A^{2}l^{2}\Xi)}}{1+A^{2}a^{2}}\,,$
		$\displaystyle V=\frac{4\pi}{3\alpha K}\left\{\frac{r_{h}(r_{h}^{2}+a^{2})}{(1-A^{2}r_{h}^{2})^{2}}+\frac{m[a^{2}(1-A^{2}l^{2}\Xi)+A^{2}l^{4}\Xi(\Xi+a^{2}l^{2})]}{(1+A^{2}a^{2})\Xi}\right\}\,,$
		$\displaystyle\lambda_{\pm}=\frac{r_{h}}{\alpha(1\pm Ar_{h})}+\frac{m}{\alpha\Xi^{2}(1+A^{2}a^{2})}\left[\Xi+\frac{a^{2}}{l^{2}}(2-A^{2}l^{2}\Xi)\right]\mp\frac{Al^{2}(\Xi+a^{2}/l^{2})}{\alpha(1+A^{2}a^{2})}\,,$
		$\displaystyle\mu_{\pm}=\frac{1}{4}\left[1-\frac{\Xi\pm 2mA}{K}\right]\,,\qquad\Xi=1-\frac{a^{2}}{l^{2}}+A^{2}a^{2}\,,\qquad P=\frac{3}{8\pi l^{2}}\,,$

It is easy to verify that the above thermodynamic quantities obey the differential first law and integral Bekenstein-Smarr mass formula simultaneously,

	$\displaystyle dM$	$\displaystyle=$	$\displaystyle TdS+\Omega dJ+VdP-\lambda_{+}d\mu_{+}-\lambda_{-}d\mu_{-}\,,$		(24)
	$\displaystyle M$	$\displaystyle=$	$\displaystyle 2TS+2\Omega\,J-2VP\,.$		(25)

In order to obtain the Gibbs free energy of the Kerr-AdS C-metric black hole, one can calculate the Euclidean action integral JHEP0517116

\mathcal{I}_{E}=\frac{1}{16\pi}\int_{M}d^{4}x\sqrt{g}\left(R+\frac{6}{l^{2}}\right)+\frac{1}{8\pi}\int_{\partial M}d^{3}x\sqrt{h}\left[\mathcal{K}-\frac{2}{l}-\frac{l}{2}\mathcal{R}(h)\right]\,,

(26)

where the extrinsic curvature $\mathcal{K}$ and the Ricci scalar $\mathcal{R}(h)$ correspond to the boundary metric $h_{\mu\nu}$ . To solve the divergence, the action includes not only the Einstein-Hilbert term but also the Gibbons-Hawking boundary term as well as the corresponding AdS boundary counterterms PRD60-104001 ; PRD60-104026 ; PRD60-104047 ; CMP208-413 ; CMP217-595 .

Using the results provided in Eq. (LABEL:ThermKerrAdSC) and the expression of the mass parameter

m=\frac{r_{h}^{2}+a^{2}}{2r_{h}}-\frac{r_{h}(r_{h}^{2}+a^{2})}{2l^{2}(A^{2}r_{h}^{2}-1)}\,,

the Gibbs free energy for the Kerr-AdS C-metric black hole is given by

G=\frac{\mathcal{I}_{E}}{\beta}=\frac{m(1-A^{2}a^{2}-2A^{2}l^{2}\Xi)}{2\alpha K(1+A^{2}a^{2})}-\frac{r_{h}(r_{h}^{2}+a^{2})}{2\alpha Kl^{2}(1-A^{2}r_{h}^{2})^{2}}=\frac{M}{2}-VP=M-TS-\Omega\,J\,.

(27)

It is easy to see that similar to the case of the Kerr C-metric black hole in the previous subsection, the conical singularity also has no effect on the calculation of the Gibbs free energy.

To determine the thermodynamic topological number of the Kerr-AdS C-metric black hole, we must first obtain the expression for the generalized off-shell Helmholtz free energy. The Helmholtz free energy is defined as

F=G+\Omega\,J=M-TS\,.

(28)

Utilizing the definition of the generalized off-shell Helmholtz free energy (1) and $l^{2}=3/(8\pi{}P)$ , one can easily calculate the result as

	$\displaystyle\mathcal{F}$	$\displaystyle=$	$\displaystyle-\frac{(r_{h}^{2}+a^{2})\sqrt{2(8\pi P-3A^{2})}(A^{2}a^{2}+1)[(8\pi P-3A^{2})r_{h}^{2}+3]\tau}{8K\tau r_{h}\sqrt{\pi P}(A^{2}r_{h}^{2}-1)[(3A^{2}-8\pi P)+3]}$		(29)
			$\displaystyle-\frac{(r_{h}^{2}+a^{2})[64\pi^{2}a^{2}Pr_{h}-24\pi(A^{2}a^{2}r_{h}+r_{h})]}{8K\tau r_{h}(A^{2}r_{h}^{2}-1)[(3A^{2}-8\pi P)a^{2}+3]}\,,$		(29)

Therefore, the components of the vector $\phi$ are computed as follows:

$\displaystyle\phi^{r_{h}}$	$\displaystyle=$	$\displaystyle\frac{\sqrt{8\pi P-3A^{2}}(1+A^{2}a^{2})}{4K\sqrt{2\pi P}r_{h}^{2}(A^{2}r_{h}^{2}-1)^{2}\big{[}(8\pi P-3A^{2})a^{2}-3\big{]}}\bigg{\{}8\pi Pr_{h}^{4}(A^{2}r_{h}^{2}-3)-3r_{h}^{2}(A^{2}r_{h}^{2}-1)^{2}$	(30)
		$\displaystyle+a^{2}\Big{[}3(A^{2}r_{h}^{2}-1)^{2}-8\pi Pr_{h}^{2}(A^{2}r_{h}^{2}+1)\Big{]}\bigg{\}}-\frac{2\pi A^{2}r_{h}(r_{h}^{2}+a^{2})}{K\tau(A^{2}r_{h}^{2}-1)^{2}}+\frac{2\pi r_{h}}{K\tau(A^{2}r_{h}^{2}-1)}\,,$	(30)
$\displaystyle\phi^{\Theta}$	$\displaystyle=$	$\displaystyle-\cot\Theta\csc\Theta\,.$	(31)

Thus the zero point of the vector field $\phi$ is

\tau=\frac{8\pi^{\frac{3}{2}}r_{h}^{3}\sqrt{2P(1+A^{2}a^{2})}\big{[}(3A^{2}-8\pi P)a^{2}+3\big{]}\big{[}(8\pi P-3A^{2})(1+A^{2}a^{2})\big{]}^{-\frac{1}{2}}}{2A^{2}r_{h}^{2}(a^{2}-r_{h}^{2})(4\pi Pr_{h}^{2}+3)-3A^{4}r_{h}^{4}(a^{2}-r_{h}^{2})+24\pi Pr_{h}^{4}+8\pi Pa^{2}r_{h}^{2}+3r_{h}^{2}-3a^{2}}\,,

(32)

which consistently reduces to the one obtained in the four-dimensional Kerr-AdS black hole case PRD107-084002 when the acceleration parameter $A$ is turned off.

For the Kerr-AdS C-metric black hole, the zero points of the component $\phi^{r_{h}}$ can be plotted with $a/r_{0}=1$ , $Ar_{0}=1$ , and $Pr_{0}^{2}=0.5$ in Fig. 4, while the unit vector field $n$ is depicted on a section of the $\Theta-r_{h}$ plane in Fig. 5 with $\tau/r_{0}=20$ . As illustrated in Fig. 4, for any given value of $\tau$ , there consistently exist two Kerr-AdS C-metric black holes: one thermodynamically stable and the other thermodynamically unstable. In Fig. 5, two zero points are located at $(r_{h}/r_{0},\Theta)=(0.33,\pi/2)$ and $(2.23,\pi/2)$ , respectively. The winding numbers $w_{i}$ for the blue contours $C_{i}$ are found to be $w_{1}=-1$ and $w_{2}=1$ , in contrast to the four-dimensional Kerr-AdS black hole, which only has $w_{1}=1$ . Consequently, the topological number $W=0$ for the Kerr-AdS C-metric black hole, as evident in Fig. 5, differs from the topological number of the four-dimensional Kerr-AdS black hole ( $W=1$ ) PRD107-084002 . This suggests that the topological number is significantly influenced by the acceleration parameter.

Furthermore, in light of another intriguing finding presented in Ref. PRD108-084041 , which demonstrate that the topological number of an non-rotating, accelerating AdS black hole is consistently lower by $-1$ compared to its non-accelerating counterpart, a broader conjecture within the framework of general relativity is proposed: the topological number of an accelerating AdS black hole consistently differs by minus one from that of its corresponding non-accelerating AdS black hole. This conjecture will be further explored in section 4.2, where we calculate the topological number of the KN-AdS C-metric black hole and compare it to that of the KN-AdS black hole, thereby affirming the validity and relevance of our proposal.

In addition, since the Kerr-AdS C-metric black hole shares the same topological number as the Kerr black hole ( $W=0$ ), we demonstrate that although the acceleration parameter and the negative cosmological constant each independently raise the topological number by one, their combined presence in the rotating black hole solution neutralizes their effects.

4 Topological classes of rotating charged accelerating black hole

In this section, we turn to explore the topological number of the four-dimensional rotating charged accelerating black hole by considering the KN C-metric solution, and then extend it to the KN-AdS C-metric case with a nonzero negative cosmological constant.

4.1 KN C-metric black hole

The metric and the Abelian gauge potential of the KN C-metric black hole are still given by Eqs. (4) and (5), however, the metric functions are respectively

	$\displaystyle f(r)$	$\displaystyle=(1-A^{2}r^{2})(r^{2}-2mr+a^{2}+q^{2})\,,$		(33)
	$\displaystyle h(\theta)$	$\displaystyle=1+2mA\cos\theta+A^{2}(a^{2}+q^{2})\cos^{2}\theta\,.$		(33)

The thermodynamic quantities are given by JHEP0419096

		$\displaystyle M=\frac{m(1-A^{2}a^{2})}{\alpha K(1+A^{2}a^{2})}\,,~{}\qquad T=\frac{\partial_{r_{h}}f(r_{h})}{4\pi\alpha(r_{h}^{2}+a^{2})}\,,~{}\qquad S=\frac{\pi(r_{h}^{2}+a^{2})}{K(1-A^{2}r_{h}^{2})}\,,$		(34)
		$\displaystyle\alpha=\frac{\sqrt{(1-A^{2}a^{2})\Xi}}{1+A^{2}a^{2}}\,,~{}\qquad J=\frac{ma}{K^{2}}\,,~{}\qquad\mu_{\pm}=\frac{1}{4}\left[1-\frac{\Xi\pm 2mA}{K}\right]\,,$
		$\displaystyle\Omega=\frac{aK}{\alpha(r_{h}^{2}+a^{2})}-\frac{A^{2}aK}{\alpha(1+A^{2}a^{2})}\,,\quad\lambda_{\pm}=\frac{r_{h}}{\alpha(1\pm Ar_{h})}-\frac{MK}{1+A^{2}a^{2}}\mp\frac{Aa^{2}}{\alpha(1+A^{2}a^{2})}\,,\quad$
		$\displaystyle Q=\frac{q}{K}\,,~{}\qquad\Phi=\frac{qr_{h}}{\alpha(r_{h}^{2}+a^{2})}\,,~{}\qquad\Xi=1+A^{2}(a^{2}+q^{2})\,.$

Nevertheless, we can verify that both the differential and integral mass formulas are completely satisfied

	$\displaystyle dM$	$\displaystyle=$	$\displaystyle TdS+\Omega dJ+\Phi dQ-\lambda_{+}d\mu_{+}-\lambda_{-}d\mu_{-}\,,$		(35)
	$\displaystyle M$	$\displaystyle=$	$\displaystyle 2TS+2\Omega\,J+\Phi Q\,.$		(36)

For the KN C-metric black hole, the expression of the Gibbs free energy can be calculated by the Euclidean action integral

\mathcal{I}_{E}=\frac{1}{16\pi}\int_{M}d^{4}x\sqrt{g}\big{(}R-F^{2}\big{)}+\frac{1}{8\pi}\int_{\partial{}M}d^{3}x\sqrt{h}(\mathcal{K}-\mathcal{K}_{0})\,.

(37)

where $h$ represents the determinant of the induced metric $h_{ij}$ , while $\mathcal{K}$ denotes the extrinsic curvature of the boundary. The term $\mathcal{K}_{0}$ indicates the subtracted value associated with the massless C-metric solution, which is utilized as the reference background. Therefore, the result of the Gibbs free energy straightforwardly reads

G=\frac{\mathcal{I}_{E}}{\beta}=\frac{m(1-A^{2}a^{2})}{2\alpha K(1+A^{2}a^{2})}-\frac{q^{2}r_{h}}{2\alpha K(r_{h}^{2}+a^{2})}=\frac{M-\Phi Q}{2}=M-TS-\Omega\,J-\Phi Q\,,

(38)

with the thermodynamic quantities in (LABEL:ThermKNC) as required. In addition, it indicate that the conical singularity also has no effect on the calculation of the Gibbs free energy in this case.

Next, we will discuss the topological number of the KN C-metric black hole. We note that the Helmholtz free energy is given by

F=G+\Omega\,J+\Phi Q=M-TS\,.

(39)

It is simple to derive the generalized off-shell Helmholtz free energy as

\mathcal{F}=M-\frac{S}{\tau}=\frac{(r_{h}^{2}+a^{2}+q^{2})(1-A^{2}a^{2})}{2\alpha Kr_{h}(1+A^{2}a^{2})}-\frac{\pi(r_{h}^{2}+a^{2})}{K\tau(1-A^{2}r_{h}^{2})}\,.

(40)

Then, the components of the vector $\phi$ are

	$\displaystyle\phi^{r_{h}}$	$\displaystyle=$	$\displaystyle\frac{(r_{h}^{2}-a^{2}-q^{2})(1-A^{2}a^{2})}{2\alpha Kr_{h}^{2}(1+A^{2}a^{2})}-\frac{2\pi r_{h}(1+A^{2}a^{2})}{K\tau(A^{2}r_{h}^{2}-1)^{2}}\,,$		(41)
	$\displaystyle\phi^{\Theta}$	$\displaystyle=$	$\displaystyle-\cot\Theta\csc\Theta\,.$		(42)

Thus, by solving the equation: $\phi^{r_{h}}$ = 0, one can easily obtain

\tau=\frac{4\pi\alpha r_{h}^{3}(1+A^{2}a^{2})^{2}}{(r_{h}^{2}-a^{2}-q^{2})(1-A^{2}a^{2})(A^{2}r_{h}^{2}-1)^{2}}

(43)

as the zero point of the vector field $\phi$ , which consistently reduces to the one obtained in the four-dimensional KN black hole PRD107-024024 when the acceleration parameter $A$ vanishes.

Considering $a/r_{0}=1$ , $q/r_{0}=1$ and $Ar_{0}=0.5$ for the KN C-metric black hole, we plot the zero points of $\phi^{r_{h}}$ in the $r_{h}-\tau$ plane in Fig. 6, and the unit vector field $n$ on a portion of the $\Theta-r_{h}$ plane with $\tau/r_{0}=20$ in Fig. 7. Evidently, for any value of $\tau$ , there exists a unique thermodynamically stable KN C-metric black hole. As shown in Fig. 7, the zero point is located at $(r_{h}/r_{0},\Theta)=(3.56,\pi/2)$ . By analyzing the local properties of this zero point, we can readily determine the topological number $W=1$ for the KN C-metric black hole. Comparing this result with the corresponding finding for the Kerr C-metric black hole in section 3.1, where $W=1$ as well, it becomes evident that the presence of the electric charge parameter does not influence the topological number of rotating, accelerating black holes. This observation further supports our initial conjecture, as proposed in Ref. PRD107-024024 , that the topological number remains unaffected by the charge in rotating black holes–a conclusion that not only extends to the case of rotating black holes in gauged supergravity theories PLB856-138919 , but also to the case of rotating, accelerating black holes. In addition, by comparing with the corresponding result for the KN black hole ( $W=0$ ), we further validate the conjecture proposed in section 3.1: the difference in topological numbers between any asymptotically flat accelerating black hole and its non-accelerating counterpart consistently equals one. This conclusion also holds true in the case of rotating, accelerating charged black holes.

4.2 KN-AdS C-metric black hole

In this subsection, we consider the general KN-AdS C-metric black hole case. The metric, the Abelian gauge potential, and the metric functions are already given in Eqs. (4)-(2). The thermodynamic quantities of the KN-AdS C-metric black hole are JHEP0419096

		$\displaystyle M=\frac{m(\Xi+a^{2}/l^{2})(1-A^{2}l^{2}\Xi)}{\Xi\alpha K(1+A^{2}a^{2})}\,,\quad T=\frac{\partial_{r_{h}}f(r_{h})}{4\pi\alpha(r_{h}^{2}+a^{2})}\,,~{}\quad S=\frac{\pi(r_{h}^{2}+a^{2})}{K(1-A^{2}r_{h}^{2})}\,,\quad Q=\frac{q}{K}\,,$		(44)
		$\displaystyle J=\frac{ma}{K^{2}}\,,\quad\Omega=\frac{aK}{\alpha(r_{h}^{2}+a^{2})}-\frac{aK(1-A^{2}l^{2}\Xi)}{l^{2}\Xi\alpha(1+A^{2}a^{2})}\,,\quad\alpha=\frac{\sqrt{(\Xi+a^{2}/l^{2})(1-A^{2}l^{2}\Xi)}}{1+A^{2}a^{2}}\,,$
		$\displaystyle V=\frac{4\pi}{3\alpha K}\left\{\frac{r_{h}(r_{h}^{2}+a^{2})}{(1-A^{2}r_{h}^{2})^{2}}+\frac{m[a^{2}(1-A^{2}l^{2}\Xi)+A^{2}l^{4}\Xi(\Xi+a^{2}l^{2})]}{(1+A^{2}a^{2})\Xi}\right\}\,,\quad\Phi=\frac{qr_{h}}{\alpha(r_{h}^{2}+a^{2})}\,,$
		$\displaystyle\lambda_{\pm}=\frac{r_{h}}{\alpha(1\pm Ar_{h})}+\frac{m}{\alpha\Xi^{2}(1+A^{2}a^{2})}\left[\Xi+\frac{a^{2}}{l^{2}}(2-A^{2}l^{2}\Xi)\right]\mp\frac{Al^{2}(\Xi+a^{2}/l^{2})}{\alpha(1+A^{2}a^{2})}\,,$
		$\displaystyle\mu_{\pm}=\frac{1}{4}\left[1-\frac{\Xi\pm 2mA}{K}\right]\,,\qquad\Xi=1-\frac{a^{2}}{l^{2}}+A^{2}(a^{2}+q^{2})\,,\qquad P=\frac{3}{8\pi l^{2}}\,,\quad$

It is simple to prove that the above thermodynamic quantities (LABEL:ThermKNAdSC) obey the first law and the Bekenstein-Smarr mass formula simultaneously

	$\displaystyle dM$	$\displaystyle=$	$\displaystyle TdS+\Omega dJ+\Phi dQ+VdP-\lambda_{+}d\mu_{+}-\lambda_{-}d\mu_{-}\,,$		(45)
	$\displaystyle M$	$\displaystyle=$	$\displaystyle 2TS+2\Omega\,J+\Phi Q-2VP\,.$		(46)

We now examine the Gibbs free energy of the KN-AdS C-metric black hole through the Euclidean action integral method. The Euclidean action is expressed as

\displaystyle\mathcal{I}_{E}=\frac{1}{16\pi}\int_{M}d^{4}x\sqrt{g}\Big{(}R+\frac{6}{l^{2}}-F^{2}\Big{)}+\frac{1}{8\pi}\int_{\partial M}d^{3}x\sqrt{h}\Big{[}\mathcal{K}-\frac{2}{l}-\frac{l}{2}\mathcal{R}(h)\Big{]}\,,

(47)

where $\mathcal{K}$ and $\mathcal{R}(h)$ denote the extrinsic curvature and the Ricci scalar of the boundary metric $h_{\mu\nu}$ , respectively. In addition to the standard Einstein-Hilbert term, the action includes the Gibbons-Hawking boundary term and the corresponding AdS boundary counterterms, which are introduced to remove divergences. Therefore, the Gibbs free energy is simply given by

	$\displaystyle G$	$\displaystyle=$	$\displaystyle\frac{\mathcal{I}_{E}}{\beta}=\frac{m(1-A^{2}a^{2}-2A^{2}l^{2}\Xi)}{2\alpha K(1+A^{2}a^{2})}-\frac{r_{h}(r_{h}^{2}+a^{2})}{2\alpha Kl^{2}(1-A^{2}r_{h}^{2})^{2}}-\frac{q^{2}r_{h}}{2\alpha K(r_{h}^{2}+a^{2})}$		(48)
		$\displaystyle=$	$\displaystyle\frac{M-\Phi Q}{2}-VP=M-TS-\Omega\,J-\Phi Q\,.$		(48)

where $\beta=T^{-1}$ represents the time coordinate’s interval. The final two equalities hold true when considering the thermodynamic variables from Eq. (LABEL:ThermKNAdSC). Moreover, it is evident that the conical singularity does not influence the calculation of the Gibbs free energy in this case.

In the following, we will investigate the topological number of the KN-AdS C-metric black hole. The Helmholtz free energy simply reads

F=G+\Omega\,J+\Phi Q=M-TS\,.

(49)

Then the generalized off-shell Helmholtz free energy is given by

	$\displaystyle\mathcal{F}$	$\displaystyle=$	$\displaystyle\frac{\sqrt{1+A^{2}(a^{2}+q^{2})}}{K[1-8\pi Pa^{2}/3+A^{2}(a^{2}+q^{2})]}\sqrt{1+\frac{A^{2}[-3+8\pi Pa^{2}-3A^{2}(a^{2}+q^{2})]}{8\pi P}}\bigg{[}\frac{r_{h}^{2}+a^{2}+q^{2}}{2r_{h}}$		(50)
			$\displaystyle+\frac{4\pi Pr_{h}(r_{h}^{2}+a^{2})}{3-3A^{2}r_{h}^{2}}\bigg{]}+\frac{\pi(r_{h}^{2}+a^{2})}{K\tau(A^{2}r_{h}^{2}-1)}\,.\quad$		(50)

Therefore the components of the vector $\phi$ can be derived as

$\displaystyle\phi^{r_{h}}$	$\displaystyle=$	$\displaystyle\frac{\sqrt{1+A^{2}(a^{2}+q^{2})}}{6K[1-8\pi Pa^{2}/3+A^{2}(a^{2}+q^{2})]}\sqrt{1+\frac{A^{2}[-3+8\pi Pa^{2}-3A^{2}(a^{2}+q^{2})]}{8\pi P}}\Bigg{\{}3-\frac{3q^{2}}{r_{h}^{2}}$	(51)
		$\displaystyle-\frac{8\pi Pr_{h}^{2}(A^{2}r_{h}^{2}-3)}{(A^{2}r_{h}^{2}-1)^{2}}+a^{2}\Bigg{[}4\pi P\bigg{(}\frac{1}{(Ar_{h}-1)^{2}}+\frac{1}{(Ar_{h}+1)^{2}}\bigg{)}\Bigg{]}\Bigg{\}}-\frac{2\pi A^{2}r_{h}(r_{h}^{2}+a^{2})}{K\tau(A^{2}r_{h}^{2}-1)^{2}}$
		$\displaystyle-\frac{2\pi r_{h}}{K\tau(1-A^{2}r_{h}^{2})}\,,$
$\displaystyle\phi^{\Theta}$	$\displaystyle=$	$\displaystyle-\cot\Theta\csc\Theta\,.$	(52)

Solving the equation $\phi^{r_{h}}=0$ straightforwardly yields

$\displaystyle\tau$	$\displaystyle=$	$\displaystyle-\frac{8\sqrt{2P}\pi^{\frac{3}{2}}r_{h}^{3}(1+A^{2}a^{2})\big{[}3-8\pi Pa^{2}+3A^{2}(a^{2}+q^{2})\big{]}}{\sqrt{\big{[}1+A^{2}(a^{2}+q^{2})\big{]}\big{[}8\pi P+A^{2}(8\pi Pa^{2}-3)-3A^{4}(a^{2}+q^{2})\big{]}}}\Big{\{}-3(r_{h}^{2}$	(53)
		$\displaystyle+8\pi Pr_{h}^{4})+3q^{2}(A^{2}r_{h}^{2}-1)^{2}+A^{2}r_{h}^{4}\big{[}r_{h}^{2}(8\pi P-3A^{2})+6\big{]}+a^{2}\big{[}3(A^{2}r_{h}^{2}-1)^{2}$
		$\displaystyle-8\pi Pr_{h}^{2}(1+A^{2}r_{h}^{2})\big{]}\Big{\}}^{-1}$

as the zero point of the vector field $\phi$ , consistent with the result for the four-dimensional KN-AdS black hole when the acceleration parameter $A$ vanishes, as detailed in Ref. PRD107-084002 .

In Figs. 8 and 9, taking $a/r_{0}=1$ , $q/r_{0}=1$ , $Ar_{0}=1$ , $Pr_{0}^{2}=0.5$ for the KN-AdS C-metric black hole, we plot the zero points of $\phi^{r_{h}}$ in the $r_{h}-\tau$ plane and the unit vector field $n$ with $\tau=20r_{0}$ , respectively. Note that for these values of $a/r_{0}$ , $q/r_{0}$ , $Ar_{0}$ and $Pr_{0}^{2}$ , there are one thermodynamically stable and one thermodynamically unstable KN-AdS C-metric black hole for any value of $\tau$ . In Fig. 9, one can observe that two zero points are located at $(r_{h}/r_{0},\Theta)=(0.42,\pi/2)$ and $(2.17,\pi/2)$ , respectively. As a result, the topological number $W=0$ for the KN-AdS C-metric black hole can be explicitly established in Figs. 8 and 9 via the local property of the zero point, which is different from that of the four-dimensional KN-AdS black hole ( $W=1$ ) PRD107-084002 . Therefore, we further verify that the conjecture proposed in section 3.2–”the topological number of an asymptotically AdS, accelerating black hole consistently differs by minus one from that of its corresponding asymptotically AdS, non-accelerating black hole”–is also applicable to the case of rotating, accelerating charged AdS black holes. Furthermore, since the KN-AdS C-metric black hole retains the same topological number as the KN black hole ( $W=0$ ), we show that although both the acceleration parameter and the negative cosmological constant individually increase the topological number by one, their combined effect in the rotating charged black hole solution counteracts this increase.

5 Concluding remarks

Table 1: The topological number

W

, numbers of generation and annihilation points for accelerating black holes and their usual nonaccelerating counterparts.

BH solution	$W$	Generation point	Annihilation point
Schwarzschild PRL129-191101	-1	0	0
Schwarzschild-AdS PRD106-064059	0	0	1
C-metric PRD108-084041	0	0	0
AdS-C-metric PRD108-084041	-1	1 or 0	1 or 0
RN PRL129-191101	0	1	0
RN-AdS PRL129-191101	1	1 or 0	1 or 0
RN-C-metric PRD108-084041	1	0	0
RN-AdS-C-metric PRD108-084041	0	1	0
Kerr PRD107-024024	0	1	0
Kerr-AdS PRD107-084002	1	1 or 0	1 or 0
Kerr C-metric	1	0	0
Kerr-AdS C-metric	0	0	0
KN PRD107-024024	0	1	0
KN-AdS PRD107-084002	1	0	0
KN C-metric	1	0	0
KN-AdS C-metric	0	0	0

Our results found in the present paper are now summarized in the following Table 1.

In this paper, we extend our prior work PRD108-084041 by employing the generalized off-shell Helmholtz free energy to explore the topological characteristics of rotating, accelerating black holes. Specifically, we analyze the topological number of the Kerr C-metric and Kerr-AdS C-metric black holes. Additionally, we apply the same approach to examine the topological number of the KN C-metric and KN-AdS C-metric black holes. Our findings reveal that the Kerr C-metric and KN C-metric black holes share the same topological classification, both having a topological number of $W=1$ , while the Kerr-AdS C-metric and KN-AdS C-metric black holes belong to a different topological class, characterized by a topological number of $W=0$ .

Building on the results presented in this paper and our previous study PRD108-084041 , we identify three intriguing outcomes:

1.

The topological number of an asymptotically flat accelerating black hole consistently differs by one from that of its non-accelerating counterpart.
2.

For an asymptotically AdS accelerating black hole, the topological number is reduced by one compared to the corresponding non-accelerating AdS black hole.
3.

In the context of general relativity, the acceleration parameter and the negative cosmological constant each independently raise the topological number by one. However, when both are present, their influences counteract, leaving the topological number unchanged.

There are two promising directions for future research emerge from this work. The first is to investigate the topological numbers of black holes in alternative theories of gravity, including scalar-tensor-vector gravity Moffat:2005si ; Liu:2023uft ; 2406.00579 ; Qiao:2024gfb , bumblebee gravity Casana:2017jkc ; Liu:2022dcn ; Liu:2024oeq , and Kalb-Ramond gravity Yang:2023wtu ; 2406.13461 ; 2407.07416 , etc. We have noted that Wei et al. 2409.09333 recently proposed a universal thermodynamic topological classification method for black hole solutions. Therefore, the second outlook is to explore the universal thermodynamic topological classes of rotating and accelerating black holes.

Acknowledgements.

This work is supported by the National Natural Science Foundation of China (NSFC) under Grants No. 12205243, No. 12375053, No. 12475051, No. 12122504, and No. 12035005; the Sichuan Science and Technology Program under Grant No. 2023NSFSC1347; the Doctoral Research Initiation Project of China West Normal University under Grant No. 21E028; the science and technology innovation Program of Hunan Province under Grant No. 2024RC1050; the innovative research group of Hunan Province under Grant No. 2024JJ1006; and the Hunan Provincial Major Sci-Tech Program under Grant No.2023ZJ1010.

References

(1) J.F. Plebanski and M. Demianski, Rotating, charged, and uniformly accelerating mass in general relativity, Ann. Phys. (N.Y.) 98 (1976) 98.
(2) W. Kinnersley and M. Walker, Uniformly accelerating charged mass in general relativity, Phys. Rev. D 2 (1970) 1359.
(3) J.B. Griffiths and J. Podolsky, A new look at the Plebanski-Demianski family of solutions, Int. J. Mod. Phys. D 15 (2006) 335 [gr-qc/0511091].
(4) W.B. Bonnor, The sources of the vacuum C-metric, Gen. Rel. Grav. 15 (1983) 535.
(5) J.B. Griffiths, P. Krtous, and J. Podolsky, Interpreting the C-metric, Class. Quant. Grav. 23 (2006) 6745 [gr-qc/0609056].
(6) T. Chen, R.-G. Cai, and B. Hu, Quasinormal modes of gravitational perturbation for uniformly accelerated black holes, Phys. Rev. D 109 (2024) 084049 [arXiv:2402.02027].
(7) R.D.B. Fontana and F.C. Mena, Quasinormal modes and stability of accelerating Reissner-Norsdtröm AdS black holes, J. High Energy Phys. 10 (2022) 047 [arXiv:2203.13933].
(8) Y. Lei, H. Shu, K. Zhang, and R.-D. Zhu, Quasinormal modes of C-metric from SCFTs, J. High Energy Phys. 02 (2024) 140 [arXiv:2308.16677].
(9) J.B. Amado and B. Gwak, Scalar quasi-normal modes of accelerating Kerr-Newman-AdS black holes, J. High Energy Phys. 02 (2024) 189 [arXiv:2309.11355].
(10) H.W. Yu, Some quantum thermal properties of uniformly accelerating and rotating black holes, Phys. Lett. A 209 (1995) 6.
(11) J.L. Zhang, Y.J. Li, and H.W. Yu, Accelerating AdS black holes as the holographic heat engines in a benchmarking scheme, Eur. Phys. J. C 78 (2018) 645 [arXiv:1801.06811].
(12) J.L. Zhang, Y.J. Li, and H.W. Yu, Thermodynamics of charged accelerating AdS black holes and holographic heat engines, J. High Energy Phys. 02 (2019) 144 [arXiv:1808.10299].
(13) J. Jiang and M. Zhang, Overcharging an accelerating Reissner-Nordström-anti-de Sitter black hole with test field and particle, Phys. Lett. B 849 (2023) 138433 [arXiv:2312.16417].
(14) M. Zhang and J. Jiang, Strong cosmic censorship in accelerating spacetime, Sci. China Phys. Mech. Astron. 66 (2023) 280412 [arXiv:2302.04738].
(15) M. Zhang and J. Jiang, Shadows of accelerating black holes, Phys. Rev. D 103 (2021) 025005 [arXiv:2010.12194].
(16) S. Jiang and J. Jiang, Holographic complexity in charged accelerating black holes, Phys. Lett. B 823 (2021) 136731 [arXiv:2106.09371].
(17) M. Zhang, C.X. Fang, and J. Jiang, Holographic complexity of rotating black holes with conical deficits, Phys. Lett. B 838 (2023) 137691 [arXiv:2212.05902].
(18) M. Astorino, Accelerating black hole in $2+1$ dimensions and $3+1$ black (st)ring, J. High Energy Phys. 01 (2011) 114 [arXiv:1101.2616].
(19) H. Lü and J.F. Vázquez-Poritz, C-metrics in gauged STU supergravity and beyond, J. High Energy Phys. 12 (2014) 057 [arXiv:1408.6531].
(20) S. Chen, M. Wang, and J. Jing, Chaotic motion of particles in the accelerating and rotating black holes spacetime, J. High Energy Phys. 09 (2016) 082 [arXiv:1604.02785].
(21) P. Ferrero, J.P. Gauntlett, J.M.P. Ipina, D. Martelli, and J. Sparks, D3-Branes Wrapped on a Spindle, Phys. Rev. Lett. 126 (2021) 111601 [arXiv:2011.10579].
(22) Pietro Ferrero, J.P. Gauntlett, and J. Sparks, Supersymmetric spindles, J. High Energy Phys. 01 (2022) 102 [arXiv:2112.01543].
(23) A. Boido, J.P. Gauntlett, D. Martelli, and J. Sparks, Entropy Functions For Accelerating Black Holes, Phys. Rev. Lett. 130 (2023) 091603 [arXiv:2210.16069].
(24) G. Arenas-Henriquez, R. Gregory, and A. Scoins, On acceleration in three dimensions, J. High Energy Phys. 05 (2022) 063 [arXiv:2202.08823].
(25) M. Astorino, Removal of conical singularities from rotating C-metrics and dual CFT entropy, J. High Energy Phys. 10 (2022) 074 [arXiv:2207.14305].
(26) G. Arenas-Henriquez, A. Cisterna, F. Diaz, and R. Gregory, Accelerating Black Holes in $2+1$ dimensions: Holography revisited, J. High Energy Phys. 09 (2023) 122 [arXiv:2308.00613].
(27) A. Cisterna, F. Diaz, R.B. Mann, and J. Oliva, Exploring accelerating hairy black holes in $2+1$ dimensions: the asymptotically locally anti-de Sitter class and its holography, J. High Energy Phys. 11 (2023) 073 [arXiv:2309.05559].
(28) J. Xu, Warped conformal symmetries of the accelerating Kerr black hole, J. High Energy Phys. 03 (2024) 050 [arXiv:2311.09831].
(29) J. Tian and T. Lai, Aspects of three-dimensional C-metric, J. High Energy Phys. 03 (2024) 079 [arXiv:2401.04457].
(30) A. Anabalon, D. Astefanesei, A. Cisterna, F. Izaurieta, J. Oliva, C. Quijada, and C. Quinzacara, Rotating and accelerating AdS black holes in Einstein-Gauss-Bonnet gravity, [arXiv:2404.04691].
(31) M. Appels, R. Gregory, and D. Kubiznak, Thermodynamics of Accelerating Black Holes, Phys. Rev. Lett. 117 (2016) 131303 [arXiv:1604.08812].
(32) M. Appels, R. Gregory, and David Kubiznak, Black hole thermodynamics with conical defects, J. High Energy Phys. 05 (2017) 116 [arXiv:1702.00490].
(33) A. Anabalon, M. Appels, R. Gregory, D. Kubiznak, R.B. Mann, and A. Ovgun, Holographic thermodynamics of accelerating black holes, Phys. Rev. D 98 (2018) 104038 [arXiv:1805.02687].
(34) A. Anabalon, F. Gray, R. Gregory, D. Kubiznak, and R.B. Mann, Thermodynamics of charged, rotating, and accelerating black holes, J. High Energy Phys. 04 (2019) 096 [arXiv:1811.04936].
(35) R. Gregory and A. Scoins, Accelerating black hole chemistry, Phys. Lett. B 796 (2019) 191 [arXiv:1904.09660].
(36) A. Ball and N. Miller, Accelerating black hole thermodynamics with boost time, Class. Quant. Grav. 38 (2021) 145031 [arXiv:2008.03682].
(37) A. Ball, Global first laws of accelerating black holes, Class. Quant. Grav. 38 (2021) 195024 [arXiv:2103.07521].
(38) D. Cassani, J.P. Gauntlett, D. Martelli, and J. Sparks, Thermodynamics of accelerating and supersymmetric AdS₄ black holes, Phys. Rev. D 104 (2021) 086005 [arXiv:2106.05571].
(39) H. Kim, N. Kim, Y. Lee, and A. Poole, Thermodynamics of accelerating AdS₄ black holes from the covariant phase space, Eur. Phys. J. C 83 (2023) 1095 [arXiv:2306.16187].
(40) C.O. Lee, Thermodynamic relation of accelerating black holes in anti-de Sitter spacetime, [arXiv:2407.21329].
(41) J.D. Bekenstein, Black holes and entropy, Phys. Rev. D 7 (1973) 2333.
(42) S.W. Hawking, Black holes and thermodynamics, Phys. Rev. D 13 (1976) 191.
(43) L. Smarr, Mass formula for Kerr black holes, Phys. Rev. Lett. 30 (1973) 71; Erratum, 30 (1973) 521.
(44) D. Christodoulou, Reversible and irreversible transforations in black hole physics, Phys. Rev. Lett. 25 (1970) 1596.
(45) D. Christodoulou and R. Ruffini, Reversible transformations of a charged black hole, Phys. Rev. D 4 (1971) 3552.
(46) D. Wu, P. Wu, H. Yu, and S.-Q. Wu, Notes on the thermodynamics of superentropic AdS black holes, Phys. Rev. D 101 (2020) 024057 [arXiv:1912.03576].
(47) D. Wu, P. Wu, H. Yu, and S.-Q. Wu, Are ultraspinning Kerr-Sen-AdS₄ black holes always superentropic?, Phys. Rev. D 102 (2020) 044007 [arXiv:2007.02224].
(48) D. Wu, S.-Q. Wu, P. Wu, and H. Yu, Aspects of the dyonic Kerr-Sen-AdS₄ black hole and its ultraspinning version, Phys. Rev. D 103 (2021) 044014 [arXiv:2010.13518].
(49) D. Wu and S.-Q. Wu, Ultra-spinning Chow’s black holes in six-dimensional gauged supergravity and their thermodynamical properties, J. High Energy Phys. 11 (2021) 031 [arXiv:2106.14218].
(50) S.-Q. Wu and D. Wu, Thermodynamical hairs of the four-dimensional Taub-Newman-Unti-Tamburino spacetimes, Phys. Rev. D 100 (2019) 101501 [arXiv:1909.07776].
(51) D. Wu and S.-Q. Wu, Consistent mass formulas for the four-dimensional dyonic NUT-charged spacetimes, Phys. Rev. D 105 (2022) 124013 [arXiv:2202.09251].
(52) D. Wu and S.-Q. Wu, Consistent mass formulas for higher even-dimensional Taub-NUT spacetimes and their AdS counterparts, Phys. Rev. D 108 (2023) 064034 [arXiv:2209.01757].
(53) S.-Q. Wu and D. Wu, Consistent mass formulas for higher even-dimensional Reissner-Nordström-NUT-AdS spacetimes, Phys. Rev. D 108 (2023) 064035 [arXiv:2306.00062].
(54) D. Wu and S.-Q. Wu, Revisiting mass formulas of the four-dimensional Reissner-Nordström-NUT-AdS solutions in a different metric form, Phys. Lett. B 846 (2023) 138227 [arXiv:2210.17504].
(55) S.-W. Wei and Y.-X. Liu, Topology of black hole thermodynamics, Phys. Rev. D 105 (2022) 104003 [arXiv:2112.01706].
(56) P.K. Yerra and C. Bhamidipati, Topology of black hole thermodynamics in Gauss-Bonnet gravity, Phys. Rev. D 105 (2022) 104053 [arXiv:2202.10288].
(57) P.K. Yerra and C. Bhamidipati, Topology of Born-Infeld AdS black holes in 4D novel Einstein-Gauss-Bonnet gravity. Phys. Lett. B 835 (2022) 137591 [arXiv:2207.10612].
(58) M.B. Ahmed, D. Kubiznak, and R.B. Mann, Vortex/anti-vortex pair creation in black hole thermodynamics, Phys. Rev. D 107 (2023) 046013 [arXiv:2207.02147].
(59) N.J. Gogoi and P. Phukon, Topology of thermodynamics in $R$ -charged black holes, Phys. Rev. D 107 (2023) 106009.
(60) M. Zhang and J. Jiang, Bulk-boundary thermodynamic equivalence: a topology viewpoint, J. High Energy Phys. 06 (2023) 115 [arXiv:2303.17515].
(61) M.R. Alipour, M.A.S. Afshar, S.N. Gashti, and J. Sadeghi, Topological classification and black hole thermodynamics, Phys. Dark Univ. 42 (2023) 101361 [arXiv:2305.05595].
(62) Z.-M. Xu, Y.-S. Wang, B. Wu, and W.-L. Yang, Riemann surface, winding number and black hole thermodynamics, Phys. Lett. B 850 (2024) 138528 [arXiv:2305.05916].
(63) M.-Y. Zhang, H. Chen, H. Hassanabadi, Z.-W. Long, and H. Yang, Topology of nonlinearly charged black hole chemistry via massive gravity, Eur. Phys. J. C 83 (2023) 773 [arXiv:2305.15674].
(64) T.N. Hung and C.H. Nam, Topology in thermodynamics of regular black strings with Kaluza-Klein reduction, Eur. Phys. J. C 83 (2023) 582 [arXiv:2305.15910].
(65) J. Sadeghi, M.R. Alipour, S.N. Gashti, and M.A.S. Afshar, Bulk-boundary and RPS thermodynamics from topology perspective, Chin. Phys. C 48 (2024) 095106 [arXiv:2306.16117].
(66) P.K. Yerra, C. Bhamidipati and S. Mukherji, Topology of critical points and Hawking-Page transition, Phys. Rev. D 106 (2022) 064059 [arXiv:2208.06388].
(67) Z.-Y. Fan, Topological interpretation for phase transitions of black holes, Phys. Rev. D 107 (2023) 044026 [arXiv:2211.12957].
(68) N.-C. Bai, L. Li and J. Tao, Topology of black hole thermodynamics in Lovelock gravity, Phys. Rev. D 107 (2023) 064015 [arXiv:2208.10177].
(69) N.-C. Bai, L. Song, and J. Tao, Reentrant phase transition in holographic thermodynamics of Born-Infeld AdS black hole, Eur. Phys. J. C 84 (2024) 43 [arXiv:2212.04341].
(70) R. Li, C.H. Liu, K. Zhang, and J. Wang, Topology of the landscape and dominant kinetic path for the thermodynamic phase transition of the charged Gauss-Bonnet AdS black holes, Phys. Rev. D 108 (2023) 044003 [arXiv:2302.06201].
(71) P.K. Yerra, C. Bhamidipati, and S. Mukherji, Topology of critical points in boundary matrix duals, J. High Energy Phys. 03 (2024) 138 [arXiv:2304.14988].
(72) Y.-Z. Du, H.-F. Li, Y.-B. Ma, and Q. Gu, Topology and phase transition for EPYM AdS black hole in thermal potential, Nucl. Phys. B 1006 (2024) 116641 [arXiv:2309.00224].
(73) P.K. Yerra, C. Bhamidipati, and S. Mukherji, Topology of Hawking-Page transition in Born-Infeld AdS black holes, J. Phys. Conf. Ser. 2667, 012031 (2023) [arXiv:2312.10784].
(74) K. Bhattacharya, K. Bamba, and D. Singleton, Topological interpretation of extremal and Davies-type phase transitions of black holes, Phys. Lett. B 854 (2024) 138722 [arXiv:2402.18791].
(75) H. Chen, M.-Y. Zhang, H. Hassanabadi, B.C. Lutfuoglu, and Z.-W. Long, Topology of dyonic AdS black holes with quasitopological electromagnetism in Einstein-Gauss-Bonnet gravity, [arXiv:2403.14730].
(76) B. Hazarika, N.J. Gogoi, and P. Phukon, Revisiting thermodynamic topology of Hawking-Page and Davies type phase transitions, [arXiv:2404.02526].
(77) T.N. Hung and C.H. Nam, Topological equivalence and phase transition rate in holographic thermodynamics of regularized Maxwell theory, [arXiv:2407.09122].
(78) I. Jeon, B.-H. Lee, W. Lee, and M. Mishra, Stability and topological nature of charged Gauss-Bonnet AdS black holes in five dimensions, [arXiv:2407.20016].
(79) H. Chen, M.Y. Zhang, A.A.A. Filho, F. Hosseinifar, and H. Hassanabadi, Thermal, topological, and scattering effects of an AdS charged black hole with an antisymmetric tensor background, [arXiv:2408.03090].
(80) J. Sadeghi, M.R. Alipour, M.A.S. Afshar, and S.N. Gashti, Exploring the phase transition in charged Gauss-Bonnet black holes: a holographic thermodynamics perspectives, Gen. Rel. Grav. 56 (2024) 93 [arXiv:2408.03126].
(81) A. Mehmood, N. Alessa, M.U. Shahzad, and E.E. Zotos, Davies-type phase transitions in 4D Dyonic AdS black holes from topological perspective, Nucl. Phys. B 1006 (2024) 116653.
(82) F. Barzi, H.E. Moumni, and K. Masmar, Riemann surfaces and winding numbers of Renyi phase structure of charged-flat black holes, [arXiv:2408.05870].
(83) P.V.P. Cunha, E. Berti, and C.A.R. Herdeiro, Light Ring Stability in Ultra-Compact Objects, Phys. Rev. Lett. 119 (2017) 251102 [arXiv:1708.04211].
(84) P.V.P. Cunha, and C.A.R. Herdeiro, Stationary Black Holes and Light Rings, Phys. Rev. Lett. 124 (2020) 181101 [arXiv:2003.06445].
(85) S.-W. Wei, Topological charge and black hole photon spheres, Phys. Rev. D 102 (2020) 064039 [arXiv:2006.02112].
(86) M. Guo and S. Gao, Universal properties of light rings for stationary axisymmetric spacetimes, Phys. Rev. D 103 (2021) 104031 [arXiv:2011.02211].
(87) R. Ghosh and S. Sarkar, Light rings of stationary spacetimes, Phys. Rev. D 104 (2021) 044019 [arXiv:2107.07370].
(88) M. Guo, Z. Zhong, J. Wang, and S. Gao, Light rings and long-lived modes in quasiblack hole spacetimes, Phys. Rev. D 105 (2022) 024049 [arXiv:2108.08967].
(89) H.C.D.L. Junior, J.-Z. Yang, L.C.B. Crispino, P.V.P. Cunha, and C.A.R. Herdeiro, Einstein-Maxwell-dilaton neutral black holes in strong magnetic fields: Topological charge, shadows, and lensing, Phys. Rev. D 105 (2022) 064070 [arXiv:2112.10802].
(90) S.-P. Wu and S.-W. Wei, Topology of light rings for extremal and nonextremal Kerr-Newman-Taub-NUT black holes without Z₂ symmetry, Phys. Rev. D 108 (2023) 104041 [arXiv:2307.14003].
(91) P.V.P. Cunha, C.A.R. Herdeiro, and J.P.A. Novo, Light rings on stationary axisymmetric spacetimes: blind to the topology and able to coexist, Phys. Rev. D 109 (2024) 064050 [arXiv:2401.05495].
(92) W. Liu, D. Wu, and J. Wang, Light rings and shadows of static black holes in effective quantum gravity, [arXiv:2408.05569].
(93) S.-W. Wei and Y.-X. Liu, Topology of equatorial timelike circular orbits around stationary black holes, Phys. Rev. D 107 (2023) 064006 [arXiv:2207.08397].
(94) X. Ye and S.-W. Wei, Topological study of equatorial timelike circular orbit for spherically symmetric (hairy) black holes, J. Cosmol. Astropart. Phys. 07 (2023) 049 [arXiv:2301.04786].
(95) X. Ye and S.-W. Wei, Novel topological phenomena of timelike circular orbits for charged test particles, [arXiv:2406.13270].
(96) S.-W. Wei, Y.-X. Liu, and R.B. Mann, Black Hole Solutions as Topological Thermodynamic Defects, Phys. Rev. Lett. 129 (2022) 191101 [arXiv:2208.01932].
(97) G.W. Gibbons and S.W. Hawking, Action integrals and partition functions in quantum gravity, Phys. Rev. D 15 (1977) 2752.
(98) J.W. York, Black-hole thermodynamics and the Euclidean Einstein action, Phys. Rev. D 33 (1986) 2092.
(99) S.-J. Yang, R. Zhou, S.W. Wei, and Y.-X. Liu, Dynamics and kinetics of phase transition for Kerr AdS black hole on free energy landscape, Phys. Rev. D 105 (2022) 084030 [arXiv:2105.00491].
(100) R. Li and J. Wang, Generalized free energy landscape of a black hole phase transition, Phys. Rev. D 106 (2022) 106015 [arXiv:2206.02623].
(101) Y.-S. Duan and M.-L. Ge, $SU$ (2) gauge theory and electrodynamics of $N$ moving magnetic monopoles, Sci. Sin. 9 (1979) 1072.
(102) Y.-S. Duan, S. Li, and G.-H. Yang, The bifurcation theory of the Gauss-Bonnet-Chern topological current and Morse function, Nucl. Phys. B 514 (1998) 705.
(103) L.-B. Fu, Y.-S. Duan, and H. Zhang, Evolution of the Chern-Simons vortices, Phys. Rev. D 61 (2000) 045004 [hep-th/0112033].
(104) C.H. Liu and J. Wang, The topological natures of the Gauss-Bonnet black hole in AdS space, Phys. Rev. D 107 (2023) 064023 [arXiv:2211.05524].
(105) C.X. Fang, J. Jiang and M. Zhang, Revisiting thermodynamic topologies of black holes, J. High Energy Phys. 01 (2023) 102 [arXiv:2211.15534].
(106) D. Wu, Topological classes of rotating black holes, Phys. Rev. D 107 (2023) 024024 [arXiv:2211.15151].
(107) D. Wu and S.-Q. Wu, Topological classes of thermodynamics of rotating AdS black holes, Phys. Rev. D 107 (2023) 084002 [arXiv:2301.03002].
(108) N. Chatzifotis, P. Dorlis, N.E. Mavromatos, and E. Papantonopoulos, Thermal stability of hairy black holes, Phys. Rev. D 107 (2023) 084053 [arXiv:2302.03980].
(109) S.-W. Wei, Y.-P. Zhang, Y.-X. Liu, and R.B. Mann, Implementing static Dyson-like spheres around spherically symmetric black hole, Phys. Rev. Res. 5, 043050 (2023) [arXiv:2303.06814].
(110) Y. Du and X. Zhang, Topological classes of black holes in de-Sitter spacetime, Eur. Phys. J. C 83 (2023) 927 [arXiv:2303.13105].
(111) C. Fairoos and T. Sharqui, Int. J. Mod. Phys. A 38 (2023) 2350133 [arXiv:2304.02889].
(112) D. Chen, Y. He, and J. Tao, Thermodynamic topology of higher-dimensional black holes in massive gravity, Eur. Phys. J. C 83 (2023) 872 [arXiv:2306.13286].
(113) N.J. Gogoi and P. Phukon, Thermodynamic topology of 4d dyonic AdS black holes in different ensembles, Phys. Rev. D 108 (2023) 066016 [arXiv:2304.05695].
(114) J. Sadeghi, S.N. Gashti, M.R. Alipour, and M.A.S. Afshar, Bardeen black hole thermodynamics from topological perspective, Ann. Phys. (Amsterdam) 455 (2023) 169391 [arXiv:2306.05692].
(115) M.S. Ali, H.E. Moumni, J. Khalloufi, and K. Masmar, Topology of Born-Infeld-AdS black hole phase transition, Ann. Phys. (Amsterdam) 465 (2024) 169679 [arXiv:2306.11212].
(116) D. Wu, Classifying topology of consistent thermodynamics of the four-dimensional neutral Lorentzian NUT-charged spacetimes, Eur. Phys. J. C 83 (2023) 365 [arXiv:2302.01100].
(117) D. Wu, Consistent thermodynamics and topological classes for the four-dimensional Lorentzian charged Taub-NUT spacetimes, Eur. Phys. J. C 83 (2023) 589 [arXiv:2306.02324].
(118) J. Sadeghi, M.A.S. Afshar, S.N. Gashti, and M.R. Alipour, Thermodynamic topology and photon spheres in the hyperscaling violating black holes, Astropart. Phys. 156 (2024) 102920 [arXiv:2307.12873].
(119) F. Barzi, H.E. Moumni, and K. Masmar, Rényi topology of charged-flat black hole: Hawking-Page and Van-der-Waals phase transitions, JHEAp 42 (2024) 63 [arXiv:2309.14069].
(120) M.U. Shahzad, A. Mehmood, S. Sharif, and A. Övgün, Criticality and topological classes of neutral Gauss-Bonnet AdS black holes in 5D, Ann. Phys. (Amsterdam) 458 (2023) 169486.
(121) C.-W. Tong, B.-H. Wang, and J.-R. Sun, Topology of black hole thermodynamics via Rényi statistics, Eur. Phys. J. C 84 (2024) 826 [arXiv:2310.09602].
(122) A. Mehmood and M.U. Shahzad, Thermodynamic topological classifications of well-known black holes, [arXiv:2310.09907].
(123) M. Rizwan and K. Jusufi, Topological classes of thermodynamics of black holes in perfect fluid dark matter background, Eur. Phys. J. C 83 (2023) 944 [arXiv:2310.15182].
(124) C. Fairoos, Topological interpretation of black hole phase transition in Gauss-Bonnet gravity, Int. J. Mod. Phys. A 39 (2024) 2450030 [arXiv:2311.04050].
(125) D. Chen, Y. He, J. Tao, and W. Yang, Topology of Hořava-Lifshitz black holes in different ensembles, Eur. Phys. J. C 84 (2024) 96 [arXiv:2311.11606].
(126) J. Sadeghi, M.A.S. Afshar, S.N. Gashti, and M.R. Alipour, Thermodynamic topology of black holes from bulk-boundary, extended, and restricted phase space perspectives, Ann. Phys. (Amsterdam) 460 (2023) 169569.
(127) B. Hazarika and P. Phukon, Thermodynamic topology of $D=4,5$ Horava Lifshitz black hole in two ensembles, Nucl. Phys. B 1006 (2024) 116649 [arXiv:2312.06324].
(128) N.J. Gogoi and P. Phukon, Thermodynamic topology of 4D Euler-Heisenberg-AdS black hole in different ensembles, Phys. Dark Univ. 44 (2024) 101456 [arXiv:2312.13577].
(129) M.-Y. Zhang, H. Chen, H. Hassanabadi, Z.-W. Long, and H. Yang, Thermodynamic topology of Kerr-Sen black holes via Rényi statistics, Phys. Lett. B 856 (2024) 138885 [arXiv:2312.12814].
(130) J. Sadeghi, M.A.S. Afshar, S.N. Gashti, and M.R. Alipour, Topology of Hayward-AdS black hole thermodynamics, Phys. Scripta 99 (2024) 025003.
(131) B. Hazarika and P. Phukon, Thermodynamic topology of black holes in f(R) gravity, [arXiv:2401.16756].
(132) D. Wu, S.-Y. Gu, X.-D. Zhu, Q.-Q. Jiang, and S.-Z. Yang, Topological classes of thermodynamics of the static multi-charge AdS black holes in gauged supergravities: novel temperature-dependent thermodynamic topological phase transition, J. High Energy Phys. 06 (2024) 213 [arXiv:2402.00106].
(133) X.-D. Zhu, D. Wu, and D. Wen, Topological classes of thermodynamics of the rotating charged AdS black holes in gauged supergravities, Phys. Lett. B 856 (2024) 138919 [arXiv:2402.15531].
(134) A. Malik, A. Mehmood, and M.U. Shahzad, Thermodynamic topological classification of higher dimensional and massive gravity black holes, Ann. Phys. (Amsterdam) 463 (2024) 169617.
(135) M.U. Shahzad, A. Mehmood, A. Malik, and A. Övgün, Topological behavior of 3D regular black hole with zero point length, Phys. Dark Univ. 44 (2024) 101437.
(136) S.-P. Wu and S.-W. Wei, Thermodynamical topology of quantum BTZ black hole, Phys. Rev. D 110 (2024) 024054 [arXiv:2403.14167].
(137) H. Chen, D. Wu, M.-Y. Zhang, H. Hassanabadi, and Z.-W. Long, Thermodynamic topology of phantom AdS black holes in massive gravity, Phys. Dark Univ. 46 (2024) 101617 [arXiv:2404.08243].
(138) B. Hazarika and P. Phukon, Topology of restricted phase space thermodynamics in Kerr-Sen-AdS black holes, [arXiv:2405.02328].
(139) Z.-Q. Chen and S.-W. Wei, Thermodynamical topology with multiple defect curves for dyonic AdS black holes, [arXiv:2405.07525].
(140) B.E. Panah, B. Hazarika, and P. Phukon, Thermodynamic topology of topological black hole in F(R)-ModMax gravity’s rainbow, [arXiv:2405.20022].
(141) H. Wang and Y.-Z. Du, Topology of the charged AdS black hole in restricted phase space, [arXiv:2406.08793].
(142) A.S. Mohamed and E.E. Zotos, Motion of test particles and topological interpretation of generic rotating regular black holes coupled to non-linear electrodynamics, Astron. Comput. 48, 100853 (2024).
(143) B. Hazarika, B.E. Panah, and P. Phukon, Thermodynamic topology of topological charged dilatonic black holes, [arXiv:2407.05325].
(144) Bi. Hazarika, A. Bhattacharjee, and P. Phukon, Thermodynamics of rotating AdS black holes in Kaniadakis statistics, [arXiv:2408.08325].
(145) Y. Sekhmani, S.N. Gashti, M.A.S. Afshar, M.R. Alipour, J. Sadeghi, and J. Rayimbaev, Thermodynamic topology of Black Holes in $F(R)$ -Euler-Heisenberg gravity’s Rainbow, [arXiv:2409.04997].
(146) M.U. Shahzad, A. Mehmood, and Ali Övgün, Thermodynamic topological classification of D-dimensional dyonic AdS black holes with quasitopological electromagnetism in Einstein-Gauss-Bonnet gravity, Eur. Phys. J. Plus 139 (2024) 806.
(147) D. Wu, Topological classes of thermodynamics of the four-dimensional static accelerating black holes, Phys. Rev. D 108 (2023) 084041 [arXiv:2307.02030].
(148) S.-Q. Wu and D. Wu, Is the type-D NUT C-metric really “missing” from the most general Plebański-Demiański solution?, [arXiv:2409.06733].
(149) S. Wang, S.-Q. Wu, F. Xie, and L. Dan, The first laws of thermodynamics of the (2+1)-dimensional BTZ black holes and Kerr-de Sitter spacetimes, Chin. Phys. Lett. 23 (2006) 1096 [hep-th/0601147].
(150) D. Kastor, S. Ray, and J. Traschen, Enthalpy and the mechanics of AdS black holes, Class. Quant. Grav. 26 (2009) 195011 [arXiv:0904.2765].
(151) M. Cvetič, G.W. Gibbons, D. Kubizňák, and C.N. Pope, Black hole enthalpy and an entropy inequality for the thermodynamic volume, Phys. Rev. D 84 (2011) 024037 [arXiv:1012.2888].
(152) R. Emparan, C.V. Johnson, and R.C. Myers, Surface terms as counterterms in the AdS/CFT correspondence, Phys. Rev. D 60 (1999) 104001 [hep-th/9903238].
(153) A. Chamblin, R. Emparan, C.V. Johnson, and R.C. Myers, Holography, thermodynamics and fluctuations of charged AdS black holes, Phys. Rev. D 60 (1999) 104026 [hep-th/9904197].
(154) R.B. Mann, Misner string entropy, Phys. Rev. D 60 (1999) 104047 [hep-th/9903229].
(155) V. Balasubramanian and P. Kraus, A Stress tensor for Anti-de Sitter gravity, Commun. Math. Phys. 208 (1999) 413 [hep-th/9902121].
(156) S.de Haro, S.N. Solodukhin, and K. Skenderis, Holographic reconstruction of space-time and renormalization in the AdS/CFT correspondence, Commun. Math. Phys. 217 (2001) 595 [hep-th/0002230].
(157) J. W. Moffat, Scalar-tensor-vector gravity theory, J. Cosmol. Astropart. Phys. 03 (2006) 004.
(158) W. Liu, X. Fang, J. Jing and J. Wang, Gravito-electromagnetic perturbations of MOG black holes with a cosmological constant: quasinormal modes and ringdown waveforms, J. Cosmol. Astropart. Phys. 11 (2023) 057.
(159) W. Liu, D. Wu, X. Fang, J. Jing, and J. Wang, Kerr-MOG-(A)dS black hole and its shadow in scalar-tensor-vector gravity theory, J. Cosmol. Astropart. Phys. 08 (2024) 035 [arXiv:2406.00579].
(160) X. Qiao, Z. W. Xia, Q. Pan, H. Guo, W. L. Qian and J. Jing, Gravitational waves from extreme mass ratio inspirals in Kerr-MOG spacetimes, [arXiv:2408.10022].
(161) R. Casana, A. Cavalcante, F. P. Poulis and E. B. Santos, Exact Schwarzschild-like solution in a bumblebee gravity model, Phys. Rev. D 97 (2018) 104001 [arXiv:1711.02273].
(162) W. Liu, X. Fang, J. Jing and J. Wang, QNMs of slowly rotating Einstein-Bumblebee black hole, Eur. Phys. J. C 83 (2023) 83 [arXiv:2211.03156].
(163) W. Liu, X. Fang, J. Jing and J. Wang, Lorentz violation induces isospectrality breaking in Einstein-bumblebee gravity theory, Sci. China Phys. Mech. Astron. 67 (2024) 280413 [arXiv:2402.09686].
(164) K. Yang, Y. Z. Chen, Z. Q. Duan and J. Y. Zhao, Static and spherically symmetric black holes in gravity with a background Kalb-Ramond field, Phys. Rev. D 108 (2023) 124004 [arXiv:2308.06613].
(165) W. Liu, D. Wu, and J. Wang, Static neutral black holes in Kalb-Ramond gravity, J. Cosmol. Astropart. Phys. 09 (2024) 017 [arXiv:2406.13461].
(166) W. Liu, D. Wu, and J. Wang, Shadow of slowly rotating Kalb-Ramond black holes, [arXiv:2407.07416].
(167) S.-W. Wei, Y.-X. Liu, and R.B. Mann, Universal topological classifications of black hole thermodynamics, [arXiv:2409.09333].