  High-temperature resummed perturbation theory is plagued by poor convergence
properties. The problem appears for theories with bosonic field content such as
QCD, QED or scalar theories. We calculate the pressure as well as other
thermodynamic quantities at high temperature for a scalar one-component field
theory, solving a three-loop 2PI effective action numerically without further
approximations. We present a detailed comparison with the two-loop
approximation. One observes a strongly improved convergence behavior as
compared to perturbative approaches. The renormalization employed in this work
extends previous prescriptions, and is sufficient to determine all counterterms
required for the theory in the symmetric as well as the spontaneously broken
phase.
