We investigate the possibility of new terms in the high-temperature axion potential arising from the dynamical nature of the axion field and from higher-order corrections to the {theta} dependence in the free energy of the quark-gluon plasma. We find that the dynamical nature of the axion field does not affect the potential but that the higher-order effects lead to new terms in the potential which are larger than the term previously considered. However, neither the magnitude nor the sign of the potential can be calculated by a perturbative expansion of the free energy since the coupling is too large. We show that a change in the magnitude of the potential does not significantly affect the bound on the axion decay constant but that the sign of the potential is of crucial importance. By investigating the formal properties of the functional integral within the instanton dilute-gas approximation, we find that the sign of the potential does not change and that the minimum remains at {theta}=0. We conclude that the standard calculation of the axion energy today is not significantly modified by this investigation.