We derive a general set of identities that determine the gauge dependence of the effective action and all associated n-point vertex functions in a wide class of gauge theories. These identities are then used to resolve the recent controversy concerning the dispersion relations in high-temperature QCD and quantum gravity. In particular, we show that the leading temperature contributions to all vertex functions in QCD, and the 1- and 2-point functions in gravity are gauge covariant and gauge fixing independent. Beyond leading order we establish gauge independence of the QCD plasma dispersion relations and show how gauge-independent dispersion relations can in principle be extracted from the thermal curved-space graviton propagator.