  Motivated by the conjecture that the cosmological constant problem is 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 spacetime volume
$V$. For the $V + V \ln V$-invariant the renormalization group running
enormously suppresses the value of the renormalized curvature which results
from Planck-size parameters specified at the Planck scale. One obtains very
large, i.e., almost flat universes without finetuning the cosmological
constant. A critical infrared fixed point is found where gravity is scale
invariant.
