  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 in the temperatures of
interest, or based on unjustifiable high-energy approximations.
