Entropy of a dressed black hole and properties of the Hartle-Hawking state

The entropy S-phi of quantum radiation in equilibrium with a black hole is obtained in a one-loop approximation without recourse to the general first law. The approach developed does not need information about quantum corrections to the Hawking temperature and metric. It follows from properties of the Hartle-Hawking state that S-phi is finite, which confirms the universality of the Bekenstein-Hawking entropy.