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We explore the possibility of a consistent cosmology based on thegauge-fixing independent running of the gravitational and cosmologicalconstants ($G$ and $\Lambda$) in the framework of effective quantum gravity. Inparticular, their running in this framework was found to satisfy $G \propto\Lambda^4$. In the cosmological setting, the covariance of the theory providesenergy conservation relations, which are impossible to satisfy with the uniquescale parameter. However, by introducing the second sub-dominant scalecorresponding to the higher-loop corrections and higher-derivative terms, onecan close the system of equations for the running of parameters and arrive atthe consistent cosmological solutions. This approach yields a change in thecosmological expansion history that affects the ratio of the Hubble parametertoday to the Hubble parameter at high redshift.

Scale-dependent cosmology from effective quantum gravity in the invariant framework