  In this work we determine how the description of a four-dimensional
Schwarzschild-anti de Sitter black hole affects the topological calculation of
its Hawking temperature. It is shown that a two-dimensional approach is
required due to the presence of the Hawking-Page phase transition which
destabilises the spacetime's topology. We prove that a dimensional reduction
removes the phase transition and hence stabilises the system. This hints at a
previously unknown feature of black hole thermodynamics, namely that certain
black holes may demand a lower-dimensional description in order to define their
Hawking temperatures.
