One-loop renormalization of black hole entropy due to nonminimally coupled matter

The quantum entanglement entropy of an eternal black hole is studied. We argue that the relevant Euclidean path integral is taken over fields defined on an {α}-fold covering of the black hole instanton. The statement that the divergences of the entropy are renormalized by the renormalization of gravitational couplings in the effective action is proved for nonminimally coupled scalar matter. The relationship of entanglement and thermodynamical entropies is discussed. {} 1995 The American Physical Society.