Phase transitions and geothermodynamics of black holes in dRGT massive gravity

In this paper, we study the thermodynamics and geothermodynamics of spherical black hole solutions in dRGT massive gravity in a new extended phase space. Inspire by the work of Kastor et al. (Class Quantum Gravity 26:195011, 2009), by interpreting the graviton mass as a thermodynamical variable, we propose a first law of thermodynamics which include a mass term and establish a new Smarr Formula. Then, we perform a thermodynamical analysis to reveal the existence of a critical behavior for black holes in dRGT massive gravity with two different critical points through canonical and grand canonical ensembles. To consolidate these results, we make use of the thermodynamical geometry formalism, with the HPEM and the Gibbs free energy metrics, to derive the singularities of Ricci scalar curvatures and show that they coincide with those of the capacities. The effect of different values of the spacetime parameters on the stability conditions is also discussed.