  In a recent paper [Franzosi, Physica A {\bf 494}, 302 (2018)], we have
suggested to use of the surface entropy, namely the logarithm of the area of a
hypersurface of constant energy in the phase space, as an expression for the
thermodynamic microcanonical entropy, in place of the standard definition
usually known as Boltzmann entropy. In the present manuscript, we have tested
the surface entropy on the Fermi-Pasta-Ulam model for which we have computed
the caloric equations that derive from both the Boltzmann entropy and the
surface entropy. The results achieved clearly show that in the case of the
Boltzmann entropy there is a strong dependence of the caloric equation from the
system size, whereas in the case of the surface entropy there is no such
dependence. We infer that the issues that one encounters when the Boltzmann
entropy is used in the statistical description of small systems could be a clue
of a deeper defect of this entropy that derives from its basic definition.
Furthermore, we show that the surface entropy is well founded from a
mathematical point of view, and we show that it is the only admissible entropy
definition, for an isolated and finite system with a given energy, which is
consistent with the postulate of equal a-priori probability.
