Abstract We elaborate on the computation of the pressure of thermal quantum electrodynamics, with massless electrons, to the fifth ( e 5) order. The calculation is performed within the Feynman gauge and the imaginary-time formalism is employed. For the e 4 calculation, the method of Sudakov decomposition is used to evaluate some ultraviolet finite integrals which have a collinear singularity. For the e 5 contribution, we give an alternative derivation and extend the discussion to massive electrons and nonzero chemical potential. Comments are made on expected similarities and differences for prospective three-loop calculations in QCD.