  A recently developed variational resummation technique, incorporating
renormalization group properties consistently, has been shown to solve the
scale dependence problem that plagues the evaluation of thermodynamical
quantities, e.g., within the framework of approximations such as in the
hard-thermal-loop resummed perturbation theory. This method is used in the
present work to evaluate thermodynamical quantities within the two-dimensional
nonlinear sigma model, which, apart from providing a technically simpler
testing ground, shares some common features with Yang-Mills theories, like
asymptotic freedom, trace anomaly and the nonperturbative generation of a mass
gap. The present application confirms that nonperturbative results can be
readily generated solely by considering the lowest-order (quasi-particle)
contribution to the thermodynamic effective potential, when this quantity is
required to be renormalization group invariant. We also show that when the
next-to-leading correction from the method is accounted for, the results
indicate convergence, apart from optimally preserving, within the
approximations here considered, the sought-after scale invariance.
