Dependence of entropy on topology? In black hole thermodynamics, there is a nontrivial dependence of thermodynamical entropy on the area of the event horizon which is a geometric object. I want to know how this may be intuitively explained (or maybe by some back-of-the-envelope calculation)? And perhaps ask, why entropy should depend on geometry and not other properties of spacetime like topology maybe?
 A: It is an interesting question. In classical general relativity the event horizon is just a virtual S2 sphere, i.e. a geometric object. However, storing information requires some material constitution of the event horizon. The dependence on topology, instead of geometry, could be the possible way to resolve this contradiction. Here I am thinking about concept of vacuole, developed by t'Hooft, https://webspace.science.uu.nl/~hooft101/lectures/GtHBlackHole_latest.pdf, or 'Hollowgraphy Driven Holography: Black Hole with Vanishing Volume Interior' by Aharon Davidson and Ily Gurwich, https://arxiv.org/abs/1007.1170, resulting in black holes without interior spacetime.
A: I don't know any 'back-of-the-envelope' type of proofs. Basically the computation of the entropy of a BH comes from the fact that at some point, there is a first order phase transition to go from 'empty space' to BH formation. This depends on the metric you are considering, i.e the metric very very far away from the BH, undisturbed by it.
I personally know only about AdS Black Hole(I remembered seeing somewhere that dS Black hole are not as stable and eventually decay but not so sure). Following a geometrical approach, one can assign a temperature to the BH, thus a free energy. Then once you have the free energy you can easily compute the entropy. For references you can have a look at: https://inspirehep.net/literature/181925 and http://www.hartmanhep.net/topics2015/gravity-lectures.pdf .
Finally, I think there is no easy intuition on what BH entropy really means... You can read the Wikipedia article http://www.scholarpedia.org/article/Bekenstein-Hawking_entropy which talks about different interpretations of this entropy
