  We study the thermodynamics and geometrothermodynamics of different black
hole configurations in more than four spacetime dimensions. We find the
conditions under which second order phase transitions occur in
higher-dimensional static Reissner-Nordstr\"om and stationary Kerr black holes.
Our results indicate that the equilibrium manifold of all these black hole
configurations is in general curved and that curvature singularities appear
exactly at those places where second order phase transitions occur.
