We study the gauge dependence of the effective average action Γk and Newtonian gravitational constant using the RG equation for Γk. Then we truncate the space of action functionals to get a solution of this equation. We solve the truncated evolution equation for the Einstein gravity in the De Sitter background for a general gauge parameter α and obtain a system of equatons for the cosmological and Newtonian constants. Analyaing the running of the gravitational constant we find that the Newtonian constant depends strongly on the gauge parameter. This leads to the appearance of antiscreening and screening behavior of the quantum gravity. The resolution of the gauge dependence problem is suggested. For physical gauges like the Landau–DeWitt gauge the Newtonian constant shows an antiscreening.