  We discuss the behavior of theories of fermions coupled to Chern-Simons gauge
fields with a non-abelian gauge group in three dimensions and at finite
temperature. Using non-perturbative arguments and gauge invariance, and in
contradiction with perturbative results, we show that the coefficient of the
Chern-Simons term of the effective actions for the gauge fields at finite
temperature can be {\it at most} an integer function of the temperature. This
is in a sense a generalized no-renormalization theorem. We also discuss the
case of abelian theories and give indications that a similar condition should
hold there too. We discuss consequences of our results to the thermodynamics of
anyon superfluids and fractional quantum Hall systems.
