Thermodynamics of de Sitter black holes with a conformally coupled scalar field

We study the thermodynamics of de Sitter black holes with a conformally coupled scalar field. The geometry is that of the lukewarm Reissner-Nordström-de Sitter black holes, with the event and cosmological horizons at the same temperature. This means that the region between the event and cosmological horizons can form a regular Euclidean instanton. The entropy is modified by the nonminimal coupling of the scalar field to the geometry, but can still be derived from the Euclidean action, provided suitable modifications are made to deal with the electrically charged case. We use the first law as derived from the isolated horizons formalism to compute the local horizon energies for the event and cosmological horizons.

Figure 1

Graph of the entropies of the event horizon, $S_{+}$ (top) and cosmological horizon, $S_{++}$ (bottom) as functions of the temperature $T$, for various values of the charge/mass ratio.

Figure 2

Graph of the cosmological horizon energy $E_{++}$ as a function of the parameter $M$ for various values of the charge/mass ratio.