  We investigate properties of the entropy density related to a generalized
extensive statistics and derive the thermodynamic Bethe ansatz equation for a
system of relativistic particles obeying such a statistics. We investigate the
conformal limit of such a system. We also derive a generalized Y-system. The
Gentile intermediate statistics and the statistics of gamma-ons are considered
in detail. In particular, we observe that certain thermodynamic quantities for
the Gentile statistics majorize those for the Haldane-Wu statistics.
Specifically, for the effective central charges related to affine Toda models
we obtain nontrivial inequalities in terms of dilogarithms.
