Cosmography of KNdS Black Holes and Isentropic Phase Transitions

We present a new analysis of Kerr-Newman-deSitter black holes in terms of thermodynamic quantities that are defined in the observable portion of the universe; between the black hole and cosmological horizons. In particular, we replace the mass $m$ with a new 'area product' parameter $X$. The physical region of parameter space is found analytically and thermodynamic quantities are given by simple algebraic functions of these parameters. We find that different geometrical properties of the black holes are usefully distinguished by the sum of the black hole and cosmological entropies. The physical parameter space breaks into a region in which the total entropy, together with $\Lambda$, $a$ and $q$ uniquely specifies the black hole, and a region in which there is a two-fold degeneracy. In this latter region, there are isentropic pairs of black holes, having the same $\Lambda$, $a$, and $q$, but different $X$. The thermodynamic volumes and masses differ in such that there are high and low density branches. The partner spacetimes are related by a simple inversion of $X$, which has a fixed point at the state of maximal total entropy. We compute the compressibility at fixed total entropy and find that it diverges at the maximal entropy point. Hence a picture emerges of high and low density phases merging at this critical point.