Entropy of a black hole with distinct surface gravities

In gravitational thermodynamics, the entropy of a black hole with distinct surface gravities can be evaluated in a microcanonical ensemble. At the WKB level, the entropy becomes the negative of the Euclidean action of the con strained instanton, which is the seed for the black hole creation in the no -boundary universe. Using the Gauss-Bonnet theorem, we prove the quite univ ersal formula in Euclidean quantum gravity that the entropy of a nonrotatin g black hole is one quarter the sum of the products of the Euler characteri stics and the areas of the horizons. For Lovelock gravity, the entropy and quantum creation of a black hole are also studied.