The renormalization group (RG) approach to cosmology is an efficient method to study the possible evolution of the cosmological parameters from the point of view of quantum field theory in curved space-time. In this work we continue our previous investigations of the RG method based on potential low-energy effects induced from physics at very high energy scales M_X near M_P. In the present instance we assume that both the Newton constant, G, and the cosmological term, \Lambda, can be functions of a scale parameter \mu. It turns out that G(\mu) evolves according to a logarithmic law which may lead to asymptotic freedom of gravity, similar to the gauge coupling in QCD. At the same time \Lambda(\mu) evolves quadratically with \mu. We study the consistency and cosmological consequences of these laws when \mu=H. Furthermore, we propose to extend this method to the astrophysical domain after identifying the local RG scale at the galactic level. It turns out that Kepler's third law of celestial mechanics receives quantum corrections that may help to explain the flat rotation curves of the galaxies without introducing the dark matter hypothesis. The origin of these effects (cosmological and astrophysical) could be linked, in our framework, to physics at M_X= 10^{; ; ; 16-17}; ; ; GeV. |