  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
$\alpha$-fold covering of the black hole instanton. The statement that
divergences of the entropy are renormalized by renormalization of gravitational
couplings in the effective action is proved for non-minimally coupled scalar
matter. The relationship of entanglement and thermodynamical entropies is
discussed.
