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Holographic Geometric Entropy at Finite Temperature from Black Holes in Global Anti de Sitter Spaces

Using a holographic proposal for the geometric entropy we study its behavior in the geometry of Schwarzschild black holes in global $AdS_p$ for $p=3,4,5$. Holographically, the entropy is determined by a minimal surface. On the gravity side, due to the presence of a horizon on the background, generically there are two solutions to the surfaces determining the entanglement entropy. In the case of $AdS_3$, the calculation reproduces precisely the geometric entropy of an interval of length $l$ in a two dimensional conformal field theory with periodic boundary conditions. We demonstrate that in the cases of $AdS_{4}$ and $AdS_{5}$ the sign of the difference of the geometric entropies changes, signaling a transition. Euclideanization implies that various embedding of the holographic surface are possible. We study some of them and find that the transitions are ubiquitous. In particular, our analysis renders a very intricate phase space, showing, for some ranges of the temperature, up to three branches. We observe a remarkable universality in the type of results we obtain from $AdS_4$ and $AdS_5$.