Entropy in the NUT–Kerr–Newman black holes due to an arbitrary spin field

Membrane method is used to compute the entropy of the NUT–Kerr–Newman black holes. It is found that even though the Euler characteristic is greater than two, the Bekenstein–Hawking area law is still satisfied. The formula S = (chi)A/8 relating the entropy and the Euler characteristic becomes inapplicable for non-extreme four-dimensional NUT–Kerr–Newman black holes.