  We consider the dynamical system given by a diagonalizable element $a$ of a
closed linear unimodular algebraic subgroup $G$ of the special linear group
over the $p$-adic numbers acting by translation on a finite volume quotient
$X$. Assuming that this action is exponentially mixing (e.g.\ if $G$ is simple)
we give an effective version (in terms of $K$-finite vectors of the regular
representation) of the following statement: If $\mu$ is an $a$-invariant
probability measure with measure-theoretical entropy close to the topological
entropy of $a$ then $\mu$ is close to the unique $G$-invariant probability
measure of $X$.
