Higher Order Corrections to the Equation of State of QED at High Temperature

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.