Gauge invariance and effective actions in D=3 at finite temperature

For background gauge field configurations reducible to the form A(mu) = ((A) over tilde(3),(A) over right arrow((x) over right arrow)) where (A) over tilde(3) is a constant, we provide an elementary derivation of the recently obtained result for the exact induced Chern-Simons (CS) effective action in QED(3) at finite temperature. The method allows us to extend the result in several useful ways: to obtain the analogous result for the "mixed" CS term in the Dorey-Mavromatos model of parity-conserving planar superconductivity, thereby justifying their argument for flux quantization in the model; to the induced CS term for a tau-dependent flux; and to the term of second order in (A) over right arrow((x) over right arrow) (and all orders in (A) over tilde(3)) in the effective action.