  At high baryon chemical potential $\mu_B$, the equation of state of QCD
allows a weak-coupling expansion in the QCD coupling $\alpha_s$. The result is
currently known up to and including the full next-to-next-to-leading order
(NNLO) $\alpha_s^2$. Starting at this order, the computations are complicated
by the modification of particle propagation in a dense medium, which
necessitates non-perturbative treatment of the scale $\alpha_s^{1/2} \mu_B$. In
this work, we apply a Hard-Thermal-Loop scheme for capturing the contributions
of this scale to the weak-coupling expansion, and use it to determine the
leading-logarithm contribution to NNNLO: $\alpha_s^3 \ln^2 \alpha_s$. This
result is the first improvement to the equation of state of massless cold quark
matter in 40 years. The new term is negligibly small, and thus significantly
increases our confidence in the applicability of the weak-coupling expansion.
