Ruppeiner theory of black hole thermodynamics
  The Ruppeiner metric as determined by the Hessian of the Gibbs surface provides a geometric description of thermodynamic systems in equilibrium. An interesting example is a black hole in equilibrium with its own Hawking radiation. In this article, we present results from the Ruppeiner study of various black hole families from different gravity theories e.g. 2D dilaton gravity, BTZ, general relativity and higher-dimensional Einstein-Maxwell gravity. 
Jan E Aman, James Bedford, Daniel Grumiller, Narit Pidokrajt and John   Ward