Haar-measure on Lie-Group has maximal entropy?

Question

Let $G$ be a Lie group and $\Gamma \subset G$ a discrete subgroup, such that there is a left-invariant probability measure $\mu$ on $G / \Gamma$. Let $g \in G$. Is it true in general, that the entropy $h_\mu (g)$ of $\mu$ w.r.t. $g$ $(\text{i.e. the map }G/\Gamma \to G/\Gamma : x \longmapsto g \cdot x)$ is maximal under all probability measures on $G / \Gamma$, i.e. $$h_\mu (g) = \sup \{ h_\nu (g) : \nu \text{ is a } g\text{-invariant probability measure on } G/\Gamma \}?$$It makes sense intuitively, seeing the Haar-measure as the equidistribution.
Where would I find such a result, including a proof? I would also be glad about statements of this sort for $G = \operatorname{SL}_n(\mathbb{R})$ and $\Gamma = \operatorname{SL}_n(\mathbb{Z})$.