Renormalized Thermodynamic Entropy of Black Holes in Higher Dimensions
  We study the ultraviolet divergent structures of the matter (scalar) field in a higher D-dimensional Reissner-Nordstr\"{o}m black hole and compute the matter field contribution to the Bekenstein-Hawking entropy by using the Pauli-Villars regularization method. We find that the matter field contribution to the black hole entropy does not, in general, yield the correct renormalization of the gravitational coupling constants. In particular we show that the matter field contribution in odd dimensions does not give the term proportional to the area of the black hole event horizon. 
Sang Pyo Kim, Sung Ku Kim, Kwang-Sup Soh, Jae Hyung Yee