Beta functions in continuum quantum gravity generally depend on the gauge and field parameterisation. The reason for this is traced to the fact that the couplings are defined as non-diffeomorphism invariant greens functions and hence are not themselves directly physical.
To recover physically meaningful beta functions I instead compute the running of a Wilsonian effective action obtained by demanding that diffeomorphism invariant observables are independent of the cut-off scale.
The one-loop running of the Newton's constant and vacuum energy are then independent of the both the gauge and field parameterisation.
In four dimensions universal quantum correction to the diffeomorphisms invariant observables are found as functions of the renormalised couplings.
These continuum results may then be used to compare to lattice studies of quantum gravity.