An important phase transition in black hole thermodynamics is associated with the divergence of the specific heat with fixed charge and angular momenta, yet one can demonstrate that neither Ruppeiner's entropy metric nor Weinhold's energy metric reveals this phase transition. In this paper, we introduce a new thermodynamical metric based on the Hessian matrix of several free energy. We demonstrate, by studying various charged and rotating black holes, that the divergence of the specific heat corresponds to the curvature singularity of this new metric. We further investigate metrics on all thermodynamical potentials generated by Legendre transformations and study correspondences between curvature singularities and phase transition signals. We show in general that for a system with n-pairs of intensive/extensive variables, all thermodynamical potential metrics can be embedded into a flat (n, n)-dimensional space. We also generalize the Ruppeiner metrics and they are all conformal to the metrics constructed from the relevant thermodynamical potentials.

It was found in [Phys. Lett.B 675 (2009) 98] that information is conserved in the process of black hole evaporation, by using the tunneling formulism and considering the correlations between emitted particles. In this Letter, we shall include quantum gravity effects, by taking into account of the log-area correction to Bekenstein-Hawking entropy. The correlation between successively emitted particles is calculated, with Planck-scale corrections. By considering the black hole evaporation process, entropy conservation is checked, and the existence of black hole remnant is emphasized. We conclude in this case information can leak out through the radiation and black hole evaporation is still a unitary process.

We study black hole phase transitions in (deformed) Hořava-Lifshitz (H-L) gravity, including the charged/uncharged topological black holes and KS black hole. Stability analysis and state space geometry are both used. We find interesting phase structures in these black holes, some of the properties are never observed in Einstein gravity. Particularly, the stability properties of black holes in H-L gravity with small radius change dramatically, which can be considered as a leak of information about the small scale behavior of spacetime. A new black hole local phase transition in H-L gravity which cannot be revealed by thermodynamical metrics has been found. There is an infinite discontinuity at the specific heat curve for charged black hole in H-L gravity with hyperbolic event horizon. However, this discontinuity does not have a corresponding curvature singularity of thermodynamical metrics. Our results may provide new insights towards a better understanding of the H-L gravity, as well as black hole thermodynamics.