The $a$-theorem and temperature of the CMB temperature in cosmology

Initial conditions in cosmology in the form of the microcanonical density matrix of the Universe predict a thermal nature of the primordial CMB power spectrum with a nonzero temperature of the resulting relict temperature distribution. This effect generates a thermal contribution to the red tilt of this spectrum, additional to its vacuum component. In the cosmological model with a large number of free fields conformally coupled to gravity the magnitude of this effect is determined by the Gauss-Bonnet coefficient \mbox{\boldmatha} of the trace anomaly. For low spins it is too small to be presently observable, but it can be amplified by the mechanism of the \mbox{\boldmatha}-theorem applied to the renormalization group flow which interpolates between the ultraviolet and infrared domains associated respectively with early and late stages of cosmological evolution.