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We consider the behavior of a quantum scalar field on three-dimensional Euclidean backgrounds: anti--de Sitter space, the regular BTZ black hole instanton, and the BTZ instanton with a conical singularity at the horizon. The corresponding heat kernel and effective action are calculated explicitly for both rotating and nonrotating holes. The quantum entropy of the BTZ black hole is calculated by differentiating the effective action with respect to the angular deficit at the conical singularity. The renormalization of the UV-divergent terms in the action and entropy is considered. The structure of the UV-finite term in the quantum entropy is of particular interest. Being negligible for large outer horizon area A+, it behaves logarithmically for small A+. Such behavior might be important at late stages of black hole evaporation.
Mann et al. (Sat,) studied this question.