Einstein-Maxwell-Dilatonic phantom black holes: Thermodynamics and geometrothermodynamics

We use the Legendre invariant formalism of geometrothermodynamics to investigate the geometric properties of the equilibrium space of a spherically symmetric phantom black hole with electric charge and dilaton. We find that at certain points of the equilibrium space, the thermodynamic curvature is characterized by the presence of singularities that are interpreted as phase transitions. We also investigate the phase transition structure by using the standard approach of black hole thermodynamics based upon the analysis of the heat capacity and response functions. We show compatibility between the two approaches. In addition, a new type of phase transition is found which is due to the presence of phantom energy and corresponds to a transition between black hole states with different stability properties.