Geometric Thermodynamics of a Schwarzschild-AdS Black Hole with a Cosmological Constant as a State Variable

The thermodynamics of the Schwarzschild-AdS black hole is reformulated within the context of the recently developed formalism of geometrothermodynamics (GTD). Different choices of the metric in the equilibrium state manifold are used in order to reproduce the Hawking-Page phase transition as a divergence of the thermodynamical curvature scalar. We show that the enthalpy and the total energy representations of GTD do not reproduce the transition while the entropy representation gives the expected behavior.