  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 $AdS_5$.
