On the ensemble dependence in black hole geometrothermodynamics

We investigate the dependence of the thermodynamic properties of black holes on the choice of statistical ensemble for a particular class of Einstein-Maxwell-Gauss-Bonnet black holes with the cosmological constant. We use partial Legendre transformations in the thermodynamic limit in order to compare the results in different ensembles, and show that the phase transition structure depends on the choice of thermodynamic potential. This result implies that thermodynamic metrics that are partially Legendre invariant cannot be employed to describe black hole thermodynamics, and partly explains why a particular thermodynamic metric has been used so far in the framework of black hole geometrothermodynamics.