The thermodynamics of the O(N) nonlinear sigma model in 1+1 dimensions is studied. We calculate the pressure to next-to-leading order in the 1/N expansion and show that at this order, only the minimum of the effective potential can be rendered finite by temperature-independent renormalization. To obtain a finite effective potential away from the minimum requires an arbitrary choice of prescription, which implies that the temperature dependence is ambiguous. We show that the problem is linked to thermal infrared renormalons. © 2004 The American Physical Society.