Recently it has been shown that the energy density contributed today by coherent oscillations of the invisible axion field corresponds to Ω ⋍ 0.1 ( ƒ a 10 12 GeV ) 11 9 ( ƒ a is the scale of peccei-Quinn SSB), apparently ruling out ƒ a ≳ O(10 13 GeV) . We propose a simple way to avoid this bound: entropy production when the temperature of the universe is between about 1 GeV and 1 MeV. The entropy might be produced by a first-order phase transition, or the out-of-equilibrium decays of a relic particle species. For ƒ a ⋍ 10 15 GeV , the entropy of the universe must be increased by a factor of O(10 2−10 3) to have Ω a ⋍ O(1) . If inflation also occurs at a temperature ⋍10 15 GeV, then the de Sitter space produced fluctuations in the axion field lead to a Zel'dovich-like spectrum of (isothermal) perturbations in the axion energy density, with amplitude about 10 −4. It has recently been argued that in an axion-dominated universe such a spectrum could lead to the formation of of the structure observed in the universe today.