  We present a finite-temperature diagrammatic perturbation theory of third
harmonic generation (THG) in doped graphene. We carry out calculations of the
third-order conductivity in the scalar potential gauge, highlighting a subtle
cancellation between a Fermi surface contribution, which contains only power
laws, and power-law contributions of inter-band nature. Only logarithms survive
in the final result. We conclude by presenting quantitative results for the
up-conversion efficiency at zero and finite temperature. Our results shed light
on the on-going dispute over the dependence of THG on carrier concentration in
graphene.
