  It is well-known that the partition function can consistently be factorized
from the canonical equilibrium distribution obtained through the maximization
of the Shannon entropy. We show that such a normalized and factorized
equilibrium distribution is warranted if and only if the entropy measure $I
\{(p_i)\}$ has an additive slope i.e. $\partial I \{(p_i)\} / \partial p_i$
when the ordinary linear averaging scheme is used. Therefore, we conclude that
the maximum entropy principle of Jaynes should not be used for the
justification of the partition functions and the concomitant thermodynamic
observables for generalized entropies with non-additive slope. Finally, Tsallis
and R\'enyi entropies are shown not to yield such factorized canonical-like
distributions.
