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We present the equation of state (pressure, trace anomaly, energy density andentropy density) of the SU(3) gauge theory from lattice field theory in anunprecedented precision and temperature range. We control both finite size andcut-off effects. The studied temperature window ($0.7... 1000 T_c$) stretchesfrom the glueball dominated system into the perturbative regime, which allowsus to discuss the range of validity of these approaches. From the criticalcouplings on fine lattices we get $T_c/\Lambdamsbar=1.26(7)$ and use this ratioto express the perturbative free energy in $T_c$ units. We also determine thepreferred renormalization scale of the Hard Thermal Loop scheme and we fit theunknown $g^6$ order perturbative coefficient at extreme high temperatures$T>100T_c$. We furthermore quantify the nonperturbative contribution to thetrace anomaly using two simple functional forms.

Lattice SU(3) thermodynamics and the onset of perturbative behaviour