  We prove that if $\mu$ is a finitely supported measure on
$\text{SL}_2(\mathbb{R})$ with positive Lyapunov exponent but not uniformly
hyperbolic, then the Lyapunov exponent function is not $\alpha$-H\"older around
$\mu$ for any $\alpha$ exceeding the Shannon entropy of $\mu$ over the Lyapunov
exponent of $\mu$.
