  Consider a statistical physical model on the $d$-regular infinite tree
$T_{d}$ described by a set of interactions $\Phi$. Let $\{G_{n}\}$ be a
sequence of finite graphs with vertex sets $V_n$ that locally converge to
$T_{d}$. From $\Phi$ one can construct a sequence of corresponding models on
the graphs $G_n$. Let $\{\mu_n\}$ be the resulting Gibbs measures. Here we
assume that $\{\mu_{n}\}$ converges to some limiting Gibbs measure $\mu$ on
$T_{d}$ in the local weak$^*$ sense, and study the consequences of this
convergence for the specific entropies $|V_n|^{-1}H(\mu_n)$. We show that the
limit supremum of $|V_n|^{-1}H(\mu_n)$ is bounded above by the
\emph{percolative entropy} $H_{perc}(\mu)$, a function of $\mu$ itself, and
that $|V_n|^{-1}H(\mu_n)$ actually converges to $H_{perc}(\mu)$ in case $\Phi$
exhibits strong spatial mixing on $T_d$. We discuss a few examples of
well-known models for which the latter result holds in the high temperature
regime.
