  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.
