Action and entropy of lukewarm black holes

We calculate the Euclidean actions and the entropies of lukewarm black holes, cold black holes and ultracold solutions in the Einstein-Maxwell theory with a positive cosmological constant. We find that the entropy of lukewarm black holes is sum of entropies of black hole horizon and cosmological horizon. The entropy of cold black holes is only one quarter area of cosmological horizon, and the entropy vanishes for ultracold solutions. We also calculate the Euler number of the corresponding manifolds, and discuss the relationship between the entropy of instanton and the Euler number. PACS numbers: 04.70.Dy, 04.20.Gz, 04.62.+v Typeset using REVTEX Electronic address: [email protected] Electronic address: [email protected] Electronic address: [email protected]