Cardy-like formula for the Schwarzschild black hole entropy

The main part of this work is to present a formula allowing a derivation of the Schwarzschild black hole entropy in arbitrary dimension. More generally, this Cardy-like formula applies for static black holes whose gravitational entropy scales as a power alpha of the temperature, and is also effective for negative heat capacity solutions alpha < 0. The formula involves the scaling power alpha, the black hole mass and the energy of a gravitational soliton. The robustness of this formula is verified in the most famous example of solution with negative heat capacity, namely the Schwarzschild black hole. The mass of the Schwarzschild regular soliton is computed using the counterterm method for asymptotically flat spacetimes. Corrections of the black hole entropy of the order of logarithm of the area are shown to arise for dimensions strictly greater than 4.