  We present preliminary results towards a fully non-perturbative determination
of the equation of state of QCD at very high-temperatures, $T\approx 3-80\,{\rm
GeV}$. The key ingredient is the lattice formulation of QCD in a moving
reference frame, which allows us for a neat determination of the entropy
density from simple expectation values of the momentum-components of the
energy-momentum tensor. For the computation we employ $N_{\rm f}=3$ flavours of
non-perturbatively O($a$)-improved Wilson-fermions. We present an analysis of
the O($a$)-improvement of the expectation values entering the determination,
and show how these can be accurately computed in simulations.
