Novel relations in massive gravity at Hawking-Page transition

Sign and magnitude of Ruppeiner's curvature RN is an empirical indicator of the respective nature and strength of microstructures of a thermodynamic system. We investigate the properties of RN for black holes in massive gravity and show that the behavior of microstructures at the HP transition point is significantly altered by the graviton mass and horizon topology. In general, depending on the massive gravity parameters, RN at HP transition point can be positive (negative) or even zero indicating respectively the dominance of repulsive (attractive) or non-interacting behaviors. In particular, presence of a weak repulsive type interaction is inferred in the large black hole branch at small pressures, which is new and in contrast to the known behavior of only attractive interactions for this branch in massless gravity. Further, when the graviton mass takes a critical value, the HP transition happens at zero temperature, where RN turns out to be a universal constant, taking exactly the same value known for Schwarzchild black hole in AdS. Analysis of the geometry of general black holes in AdS close to the HP transition point via novel near horizon scaling limits, reveals the presence of fully decoupled Rindler spacetimes.