Consider the heat equation on (real) hyperbolic space $\mathbb{H}^n$ with initial data $f$. It is well-known that under mild conditions on $f$, the solution converges pointwise a.e. to $f$ as time goes to zero. For rougher initial data, we characterize the weights $v$ on $\mathbb{H}^n$ for which the solution converges pointwise a.e. to the initial data when the latter is in $L^p(v)$, $1 \leq p < \infty$. As a tool, we also establish vector-valued weak type $(1, 1)$ and $L^p$ estimates ($1 < p < \infty$) for the local Hardy–Littlewood maximal function on $\mathbb{H}^n$. Our results hold on arbitrary rank symmetric spaces and for alternative versions of the Laplacian (shifted, distinguished), as well as for the fractional heat equation and the Caffarelli-Silvestre extension.