We are interested in the Guivarc'h inequality for admissible random walks on
finitely generated relatively hyperbolic groups, endowed with a word metric. We
show that for random walks with finite super-exponential moment, if this
inequality is an equality, then the Green distance is roughly similar to the
word distance, generalizing results of Blach{\`e}re, Ha{\"i}ssinsky and Mathieu
for hyperbolic groups [4]. Our main application is for relatively hyperbolic
groups with respect to virtually abelian subgroups of rank at least 2. We show
that for such groups, the Guivarc'h inequality with respect to a word distance
and a finitely supported random walk is always strict
