  We investigate the thermodynamic properties of a novel class of gauge-Yukawa
theories that have recently been shown to be completely asymptotically safe,
because their short-distance behaviour is determined by the presence of an
interacting fixed point. Not only do all the coupling constants freeze at a
constant and calculable value in the ultraviolet, their values can even be made
arbitrarily small for an appropriate choice of the ratio $N_c/N_f$ of fermion
colours and flavours in the Veneziano limit. Thus, a perturbative treatment can
be justified. We compute the pressure, entropy density, and thermal degrees of
freedom of these theories to next-to-next-to-leading order in the coupling
constants.
