  We show that in any model of non-eternal inflation satisfying the null energy
condition, the area of the de Sitter horizon increases by at least one Planck
unit in each inflationary e-folding. This observation gives an operational
meaning to the finiteness of the entropy S_dS of an inflationary de Sitter
space eventually exiting into an asymptotically flat region: the asymptotic
observer is never able to measure more than e^(S_dS) independent inflationary
modes. This suggests a limitation on the amount of de Sitter space outside the
horizon that can be consistently described at the semiclassical level, fitting
well with other examples of the breakdown of locality in quantum gravity, such
as in black hole evaporation. The bound does not hold in models of inflation
that violate the null energy condition, such as ghost inflation. This
strengthens the case for the thermodynamical interpretation of the bound as
conventional black hole thermodynamics also fails in these models, strongly
suggesting that these theories are incompatible with basic gravitational
principles.
