  In 2+1 dimensions, QED becomes exactly solvable for all values of the fermion
charge $e$ in the limit of many fermions $N_f\gg 1$. We present results for the
free energy density at finite temperature $T$ to next-to-leading-order in large
$N_f$. In the naive large $N_f$ limit, we uncover an apparently UV-divergent
contribution to the vacuum energy at order ${\cal O}(e^6 N_f^3)$, which we
argue to become a finite contribution of order ${\cal O}(N_f^4 e^6)$ when
resumming formally higher-order $1/N_f$ contributions. We find the
finite-temperature free energy to be well-behaved for all values of the
dimensionless coupling $e^2N_f/T$, and to be bounded by the free energy of
$N_f$ free fermions and non-interacting QED3, respectively. We invite follow-up
studies from finite-temperature lattice gauge theory at large but fixed $N_f$
to test our results in the regime $e^2N_f/T\gg 1$.
