  For cosmologies including scale dependence of both the cosmological and the
gravitational constant, an additional consistency condition dictated by the
Bianchi identities emerges, even if the energy-momentum tensor of ordinary
matter stays individually conserved. For renormalization-group (RG) approaches
it is shown that such a consistency relation ineluctably fixes the RG scale
(which may have an explicit as well as an implicit time dependence), provided
that the solutions of the RG equation for both quantities are known. Hence,
contrary to the procedures employed in the recent literature, we argue that
there is no more freedom in identification of the RG scale in terms of the
cosmic time in such cosmologies. We carefully set the RG scale for the RG
evolution phrased in a quantum gravity framework based on the hypothetical
existence of an infrared (IR) fixed point, for the perturbative regime within
the same framework, as well as for an evolution within quantum field theory
(QFT) in a curved background. In the latter case, the implications of the scale
setting for the particle spectrum are also briefly discussed.
