Thermodynamics of Regular Cosmological Black Holes with the de Sitter Interior

We address the question of thermodynamics of regular cosmological spherically symmetric black holes with the de Sitter center. Space-time is asymptotically de Sitter as r -> 0 and as r -> infinity. A source term in the Einstein equations connects smoothly two de Sitter vacua with different values of cosmological constant: 8 pi GT(nu)(mu) = Lambda delta(mu)(nu) as r -> 0, 8 pi GT(nu)(mu) = lambda delta(mu)(nu) as r -> infinity with lambda < Lambda. It represents an anisotropic vacuum dark fluid defined by symmetry of its stress-energy tensor which is invariant under the radial boosts. In the range of the mass parameter M-cr1 <= M <= M-cr2 it describes a regular cosmological black hole. Space-time in this case has three horizons: a cosmological horizon r(c), a black hole horizon r(b) < r(c), and an internal horizon r(a) < r(b), which is the cosmological horizon for an observer in the internal R-region asymptotically de Sitter as r -> 0. We present the basic features of space-time geometry and the detailed analysis of thermodynamics of horizons using the Padmanabhan approach relevant for a multi-horizon space-time with a non-zero pressure. We find that in a certain range of parameters M and q = root Lambda/lambda there exist a global temperature for an observer in the R-region between the black hole horizon r(b) and cosmological horizon r(c). We show that a second-order phase transition occurs in the course of evaporation, where a specific heat is broken and a temperature achieves its maximal value. Thermodynamical preference for a final point of evaporation is thermodynamically stable double-horizon (r(a) = r(b)) remnant with the positive specific heat and zero temperature.