Thermodynamics of phantom black holes in Einstein-Maxwell-dilaton theory

A thermodynamic analysis of the black hole solutions coming from the Einstein-Maxwell-dilaton theory in 4D is done. By considering the canonical and grand-canonical ensemble, we apply standard method as well as a recent method known as geometrothermodynamics. We are particularly interested in the characteristics of the so called phantom black hole solutions. We will analyze the thermodynamics of these solutions, the points of phase transition and their extremal limit. The thermodynamic stability is also analyzed. We obtain a mismatch between the results of the geometrothermodynamics method when compared with the ones obtained by the specific heat, revealing a weakness of the method, as well as possible limitations of its applicability to very pathological thermodynamic systems. We also found that normal and phantom solutions are locally and globally unstable, except for certain values of the coupled constant of the Einstein-Maxwell-dilaton action. We also show that the anti-Reissner-Nordstrom solution does not possess extremal limit nor phase transition points, contrary to the Reissner-Nordstrom case.