  We calculate the pressure of massless $\phi^4$-theory to order $g^8\log(g)$
at weak coupling. The contributions to the pressure arise from the hard
momentum scale of order $T$ and the soft momentum scale of order $gT$.
Effective field theory methods and dimensional reduction are used to separate
the contributions from the two momentum scales: The hard contribution can be
calculated as a power series in $g^2$ using naive perturbation theory with bare
propagators. The soft contribution can be calculated using an effective theory
in three dimensions, whose coefficients are power series in $g^2$. This
contribution is a power series in $g$ starting at order $g^3$. The calculation
of the hard part to order $g^6$ involves a complicated four-loop sum-integral
that was recently calculated by Gynther, Laine, Schr\"oder, Torrero, and
Vuorinen. The calculation of the soft part requires calculating the mass
parameter in the effective theory to order $g^6$ and the evaluation of
five-loop vacuum diagrams in three dimensions. This gives the free energy
correct up to order $g^7$. The coefficients of the effective theory satisfy a
set of renormalization group equations that can be used to sum up leading and
subleading logarithms of $T/gT$. We use the solutions to these equations to
obtain a result for the free energy which is correct to order $g^8\log(g)$.
Finally, we investigate the convergence of the perturbative series.
