Geometry of criticality, supercriticality, and Hawking-Page transitions in Gauss-Bonnet-AdS black holes

We obtain the Ruppeiner geometry associated with the nonextended state space (Λ constant) of the charged Gauss-Bonnet AdS (GB-AdS) black holes and confirm that the state space Riemannian manifold becomes strongly curved in regions where the black hole system develops strong statistical correlations in the grand canonical ensemble (M and Q fluctuating). We establish the exact proportionality between the state space scalar curvature R and the inverse of the singular free energy near the isolated critical point for the grand canonical ensemble in spacetime dimension $d=5$, thus hopefully moving a step closer to the agenda of a concrete physical interpretation of R for black holes. On the other hand, we show that while R signals the Davies transition points (which exist in GB-AdS black holes for d≥6) through its divergence, it does not scale as the inverse of the singular free energy there. Furthermore, adapting to the black hole case the ideas developed in [1] in the context of pure fluids, we find that the state space geometry encodes phase coexistence and first order transitions, identifies the asymptotically critical region and even suggests a Widom-line-like crossover regime in the supercritical region for 5-d case. The sign of R appears to imply a significant difference between the microscopic structure of the small and the large black hole branches in $d=5$. We show that thermodynamic geometry informs the microscopic nature of coexisting thermal GB-AdS and black hole phases near the Hawking-Page phase transition.