It is shown that curvature effects, though completely negligible at the present day, can drastically alter thermal spectra at early times in some inflationary cosmological models. This can lead to a modification of the thermal history of the universe before the entropy production. The effects are only present for spatially curved k not=0 Robertson-Walker models, not for spatially flat k=0 models. If the entropy increase is >> 1087 and k not=0 then the energy density in the quantum fields is much greater, and hence the universe is much younger, at the onset of inflation than would be the case in a spatially flat universe. The effect is not relevant for chaotic inflation.