The thermodynamics of the O(N) linear and nonlinear sigma models in 3 + 1 dimensions is studied. We calculate the pressure to next-to-leading order in the 1/N expansion and show that at this order, temperature-independent renormalization is only possible at the minimum of the effective potential. The 1/N expansion is found to be a good expansion for N as low as 4, which is the case relevant for low-energy QCD phenomenology. We consider the cases with and without explicit symmetry breaking. We show that previous next-to-leading-order calculations of the pressure are either breaking down at the temperatures of interest, or based on unjustifiable high-energy approximations.