  Spacetime wormholes can provide non-perturbative contributions to the
gravitational path integral that make the actual number of states $e^S$ in a
gravitational system much smaller than the number of states
$e^{S_{\mathrm{p}}}$ predicted by perturbative semiclassical effective field
theory. The effects on the physics of the system are naturally profound in
contexts in which the perturbative description actively involves $N = O(e^S)$
of the possible $e^{S_{\mathrm{p}}}$ perturbative states; e.g., in late stages
of black hole evaporation. Such contexts are typically associated with the
existence of non-trivial quantum extremal surfaces. However, by forcing a
simple topological gravity model to evolve in time, we find that such effects
can also have large impact for $N\ll e^S$ (in which case no quantum extremal
surfaces can arise). In particular, even for small $N$, the insertion of
generic operators into the path integral can cause the non-perturbative time
evolution to differ dramatically from perturbative expectations. On the other
hand, this discrepancy is small for the special case where the inserted
operators are non-trivial only in a subspace of dimension $D \ll e^S$. We thus
study this latter case in detail. We also discuss potential implications for
more realistic gravitational systems.
