A Feynman-Jensen version of the thermal variational principle is applied to hot gauge fields, Abelian as well as non-Abelian: scalar electrodynamics (without scalar self-coupling) and the gluon plasma. The perturbatively known self-energies are shown to derive by variation from a free quadratic (''Gaussian'') trial Lagrangian. Independence of the covariant gauge fixing parameter is reached (within the order $g^3$ studied) after a reformulation of the partition function such that it depends on only even powers of the gauge field. Also static properties (Debye screening) are reproduced this way. But because of the present need to expand the variational functional, the method falls short of its potential nonperturbative power.

36 pages, LaTeX, no figures. Updated version: new title, section on static properties and some references added