  We investigate the renormalization group optimized perturbation theory
(RGOPT) at the next-to-next-to-leading order (NNLO) for the thermal scalar
field theory. From comparing three thus available successive RGOPT orders we
illustrate the efficient resummation and very good apparent convergence
properties of the method. In particular the remnant renormalization scale
dependence of thermodynamical quantities is drastically improved as compared to
both standard perturbative expansions and other related resummation methods,
such as the screened perturbation theory. Our present results thus constitute a
useful first NNLO illustration in view of NNLO applications of this approach to
the more involved thermal QCD.

High Energy Physics - Phenomenology