Consider a statistical physical model on the $d$-regular infinite tree $T_{d}$ described by a set of interactions $Φ$. Let $\{G_{n}\}$ be a sequence of finite graphs with vertex sets $V_n$ that locally converge to $T_{d}$. From $Φ$ one can construct a sequence of corresponding models on the graphs $G_n$. Let $\{μ_n\}$ be the resulting Gibbs measures. Here we assume that $\{μ_{n}\}$ converges to some limiting Gibbs measure $μ$ on $T_{d}$ in the local weak$^*$ sense, and study the consequences of this convergence for the specific entropies $|V_n|^{-1}H(μ_n)$. We show that the limit supremum of $|V_n|^{-1}H(μ_n)$ is bounded above by the \emph{percolative entropy} $H_{perc}(μ)$, a function of $μ$ itself, and that $|V_n|^{-1}H(μ_n)$ actually converges to $H_{perc}(μ)$ in case $Φ$ 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.