Black hole entropy: Off shell versus on shell

Different methods of calculation of quantum corrections to the thermodynamical characteristics of a black hole are compared. The general relation between on-shell and off-shell approaches is discussed. We consider the simplified 2D model with the dilaton gravity and show in its framework that the observable thermodynamical black hole entropy can be presented in the form S^TD=r+^2+S^SM-Sₑ₈₍₃₋₄ₑ^SM. Here, r+ is the radius of the horizon shifted because of the quantum back reaction effect, S^SM is the statistical-mechanical entropy, determined as S^SM=-Tr (^{}^Hln^{}^H) for the density matrix ^{}^H of a black hole, and Sₑ₈₍₃₋₄ₑ^SM is the analogous entropy in the Rindler space.