Calculate $h_\mu $ for $G/\Gamma $ finite measure, non compact.
When $G$ is of rank one, there is the article of M. Peigné and J.P. Otal  for the measure of maximal entropy, and the book of M. Pollicott,
F. Paulin and B. Schapira  for general Gibbs measures... and it works
 MR2097356, Duke Math. J. 125 (2004)
 MR3444431, Astérisque No. 373 (2015)