The QCD Equation of State with $N_f=3$ massless quark flavours is determined non-perturbatively over a broad range of
temperatures, extending from the electroweak scale down to 3 GeV, and smoothly connecting to the low-temperature regime.
The comparison with perturbative predictions shows that, even at temperatures approaching the electroweak scale, the Equation of State can
be accurately described only by adding terms beyond the known perturbative series, including non-perturbative contributions.
The strategy that allows this investigation in the previously unexplored high-temperature regime combines shifted boundary conditions
with a determination of the lines of constant physics based on the running of a non-perturbatively defined renormalized coupling. This
methodology is general and can be applied to QCD with four or five massive quark flavours.

