Black hole entropy: thermodynamics, statistical-mechanics and subtraction procedure

The thermodynamical one-loop entropy STD of a two-dimensional black hole in thermal equilibrium with the massless quantum gas is calculated. It is shown that STD includes the Bekenstein-Hawking entropy, evaluated for the quantum corrected geometry, and the finite difference of statistical mechanical entropies — Tr ϱ ln ϱ for the gas on the black hole and Rindler spaces. This result demonstrates in an explicit form that the relation between thermodynamical and statistical-mechanical entropies of a black hole is non-trivial and requires special subtraction procedure.