Flat Information Geometries in Black Hole Thermodynamics
  The Hessian of either the entropy or the energy function can be regarded as a metric on a Gibbs surface. For two parameter families of asymptotically flat black holes in arbitrary dimension one or the other of these metrics are flat, and the state space is a flat wedge. The mathematical reason for this is traced back to the scale invariance of the Einstein-Maxwell equations. The picture of state space that we obtain makes some properties such as the occurence of divergent specific heats transparent. 
J. E. Aman, Ingemar Bengtsson and Narit Pidokrajt