  Persistent puzzles to do with information loss for black holes have
stimulated critical reassessment of the domain of validity of semiclassical EFT
reasoning in curved spacetimes, particularly in the presence of horizons. We
argue here that perturbative predictions about evolution for very long times
near a horizon are subject to problems of secular growth - i.e. powers of small
couplings come systematically together with growing functions of time. Such
growth signals a breakdown of naive perturbative calculations of late-time
behaviour, regardless of how small ambient curvatures might be. Similar issues
of secular growth also arise in cosmology, and we build evidence for the case
that such effects should be generic for gravitational fields. In particular,
inferences using free fields coupled only to background metrics can be
misleading at very late times due to the implicit assumption they make of
perturbation theory when neglecting other interactions. Using the Rindler
horizon as an example we show how this secular growth parallels similar
phenomena for thermal systems, and how it can be resummed to allow late-time
inferences to be drawn more robustly. Some comments are made about the
appearance of an IR/UV interplay in this calculation, as well as on the
possible relevance of our calculations to predictions near black-hole horizons.
