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The equation of state of the SU($3$) Yang-Mills theory is determined in thedeconfined phase with a precision of about 0.5%. The calculation is carried outby numerical simulations of lattice gauge theory with shifted boundaryconditions in the time direction. At each given temperature, up to $230\, T_c$with $T_c$ being the critical temperature, the entropy density is computed atseveral lattice spacings so to be able to extrapolate the results to thecontinuum limit with confidence. Taken at face value, above a few $T_c$ theresults exhibit a striking linear behaviour in $\ln(T/T_c)^{-1}$ over almost 2orders of magnitude. Within errors, data point straight to the Stefan-Boltzmannvalue but with a slope grossly different from the leading-order perturbativeprediction. The pressure is determined by integrating the entropy in thetemperature, while the energy density is extracted from $T s=(\epsilon + p )$.The continuum values of the potentials are well represented by Pad\'einterpolating formulas, which also connect them well to the Stefan-Boltzmannvalues in the infinite temperature limit. The pressure, the energy and theentropy densities are compared with results in the literature. The discrepancyamong previous computations near $T_c$ is analyzed and resolved thanks to thehigh precision achieved.

Equation of state of the SU($3$) Yang-Mills theory: a precise determination from a moving frame