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.