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In this study,we proposed two anharmonic phonon theories for momentum conserving nonlinear lattices based on a generalized Gibbs-Bogoliubov inequality.This inequality provides a upper bound as well as a lower bound on the Gibbs free energy.Using the Gibbs free energy instead of the Helmholtz free energy enables us to investigate systems under nonzero pressure.We demonstrate the power of our theories by considering applications to one dimensional nonlinear lattices with a symmetric or asymmetric Fermi-Pasta-Ulam potential.Among two anharmonic phonon theories,we found the theory derived from the lower bound of the generalized Gibbs-bogoliubov inequality gives better theoretical predictions.Excellent agreements with molecule dynamics results bear out the superiority of our theory compared with existing quasiharmonic theories.