Topological classes of thermodynamics of the four-dimensional static accelerating black holes

Abstract: In this paper, utilizing the generalized off shell Helmholtz free energy, we explore the topological numbers of the four-dimensional static accelerating black hole and its AdS extension, as well as the static charged accelerating black hole and its AdS extension. Our analysis reveals a profound and significant impact of the acceleration parameter on the topological numbers associated with the static black holes; and different values (nonzero) of the acceleration parameter do not affect the topological numbers of the corresponding four-dimensional static accelerating black holes. In addition, we demonstrate that the electric charge parameter has an important effect on the topological number of the static neutral accelerating black holes, and the cosmological constant has a remarkable influence on the topological number of the static accelerating black hole. Furthermore, it is interesting to observe that the difference between the topological number of the asymptotically flat static accelerating black hole and that of its corresponding asymptotically flat static nonaccelerating black hole is always unity, and the difference between the topological number of the asymptotically AdS static accelerating black hole and that of its corresponding asymptotically AdS static nonaccelerating black hole is always $-1$. This new observation leads us to conjure that it might be valid also for other accelerating black holes. Of course, this captivating conjecture requires empirical verification through comprehensive investigation into the topological numbers of other accelerating black holes and their corresponding usual counterparts.