  We prove that prethermalization is a generic property of gapped local
many-body quantum systems, subjected to small perturbations, in any spatial
dimension. More precisely, let $H_0$ be a Hamiltonian, spatially local in $d$
spatial dimensions, with a gap $\Delta$ in the many-body spectrum; let $V$ be a
spatially local Hamiltonian consisting of a sum of local terms, each of which
is bounded by $\epsilon \ll \Delta$. Then, the approximation that quantum
dynamics is restricted to the low-energy subspace of $H_0$ is accurate, in the
correlation functions of local operators, for stretched exponential time scale
$\tau \sim \exp[(\Delta/\epsilon)^a]$ for any $a<1/(2d-1)$. This result does
not depend on whether the perturbation closes the gap. It significantly extends
previous rigorous results on prethermalization in models where $H_0$ was
frustration-free. We infer the robustness of quantum simulation in low-energy
subspaces, the existence of athermal ``scarred" correlation functions in gapped
systems subject to generic perturbations, the long lifetime of false vacua in
symmetry broken systems, and the robustness of quantum information in
non-frustration-free gapped phases with topological order.
