Renormalized Thermodynamic Entropy of Black Holes

We study the thermodynamic entropy of a massive scalar field in the spacetime of the two-dimensional black hole of the Jackiw-Teitelboim gravity, and a $D$-dimensional Reissner-Nordström charged black hole. By regularizing the ultraviolet divergences using the Pauli-Villars regularization method, we obtain the renormalized black hole entropy of the scalar field which has a contribution proportional to the black hole area only in an even dimensional black hole but not in an odd dimensional black hole, implying that the quantum effect of matter does not renormalize the black hole entropy in the odd dimension. We obtain explicitly the renormalized thermodynamic entropy in five- and six-dimensional black holes