  A recently developed variant of the so-called optimized perturbation theory
(OPT), making it perturbatively consistent with renormalization group (RG)
properties, RGOPT, was shown to drastically improve its convergence for zero
temperature theories. Here the RGOPT adapted to finite temperature is
illustrated with a detailed evaluation of the two-loop pressure for the thermal
scalar $ \lambda\phi^4$ field theory. We show that already at the simple
one-loop level this quantity is exactly scale-invariant by construction and
turns out to qualitatively reproduce, with a rather simple procedure, results
from more sophisticated resummation methods at two-loop order, such as the
two-particle irreducible approach typically. This lowest order also reproduces
the exact large-$N$ results of the $O(N)$ model. Although very close in spirit,
our RGOPT method and corresponding results differ drastically from similar
variational approaches, such as the screened perturbation theory or its
QCD-version, the (resummed) hard thermal loop perturbation theory. The latter
approaches exhibit a sensibly degrading scale dependence at higher orders,
which we identify as a consequence of missing RG invariance. In contrast RGOPT
gives a considerably reduced scale dependence at two-loop level, even for
relatively large coupling values $\sqrt{\lambda/24}\sim {\cal O}(1)$, making
results much more stable as compared with standard perturbation theory, with
expected similar properties for thermal QCD.
