  We present the equation of state (pressure, trace anomaly, energy density and
entropy density) of the SU(3) gauge theory from lattice field theory in an
unprecedented precision and temperature range. We control both finite size and
cut-off effects. The studied temperature window (0.7...1000 T_c) stretches from
the glueball dominated system into the perturbative regime, which allows us to
discuss the range of validity of these approaches. We also determine the
preferred renormalization scale of the Hard Thermal Loop scheme and we fit the
unknown g^6 order perturbative coefficient at extreme high temperatures T>100
T_c. We furthermore quantify the nonperturbative contribution to the trace
anomaly using a simple functional form. Our high precision data allows one to
have a complete theoretical description of the equation of state from T=0 all
the way to the phase transition, through the transition region into the
perturbative regime up to the Stefan-Boltzmann limit. We will discuss this
description, too.
