  We analyze the conceptual role of background independence in the application
of the effective average action to quantum gravity. Insisting on a background
independent renormalization group (RG) flow the coarse graining operation must
be defined in terms of an unspecified variable metric since no rigid metric of
a fixed background spacetime is available. This leads to an extra field
dependence in the functional RG equation and a significantly different RG flow
in comparison to the standard flow equation with a rigid metric in the mode
cutoff. The background independent RG flow can possess a non-Gaussian fixed
point, for instance, even though the corresponding standard one does not. We
demonstrate the importance of this universal, essentially kinematical effect by
computing the RG flow of Quantum Einstein Gravity in the ``conformally
reduced'' Einstein--Hilbert approximation which discards all degrees of freedom
contained in the metric except the conformal one. Without the extra field
dependence the resulting RG flow is that of a simple $\phi^4$-theory. Including
it one obtains a flow with exactly the same qualitative properties as in the
full Einstein--Hilbert truncation. In particular it possesses the non-Gaussian
fixed point which is necessary for asymptotic safety.
