  Motivated by the conjecture that the cosmological constant problem could be
solved by strong quantum effects in the infrared we use the exact flow equation
of Quantum Einstein Gravity to determine the renormalization group behavior of
a class of nonlocal effective actions. They consist of the Einstein-Hilbert
term and a general nonlinear function F(k, V) of the Euclidean space-time
volume V. A partial differential equation governing its dependence on the scale
k is derived and its fixed point is analyzed. For the more restrictive
truncation of theory space where F(k, V) is of the form V+V ln V, V+V^2, and
V+\sqrt{V}, respectively, the renormalization group equations for the running
couplings are solved numerically. The results are used in order to determine
the k-dependent curvature of the S^4-type Euclidean space-times which are
solutions to the effective Einstein equations, i.e. stationary points of the
scale dependent effective action. For the V+V ln V-invariant (discussed earlier
by Taylor and Veneziano) we find that the renormalization group running
enormously suppresses the value of the renormalized curvature which results
from Planck-size bare parameters specified at the Planck scale. Hence one can
obtain very large, almost flat universes without fine-tuning the cosmological
constant.
