Cosmology with modified continuity equation in asymptotically safe gravity

We study FLRW cosmology, taking into account quantum gravitational corrections in the formalism of the exact renormalization group flow of the effective action for gravity. We calculate the quantum corrected scale factor, energy density, and entropy production at late times, taking different cut-off functions. Our approach differs from previous ones in the way energy-momentum conservation is imposed -- we include the running Newton constant $G(k)$ in the definition of energy-momentum tensor, keeping in mind the covariant conservation identity of the Einstein tensor. The quantum corrections obtained in this approach are different from what are found by letting the conservation equation remain the same as for a scale-independent Newton constant. We also find that for a specific choice of the cut-off scale, the quantum corrected behaviour of the Newton constant and the cosmological constant lead to an emergent universe solution.