  We use the Wetterich-equation to study the renormalization group flow of
$f(R)$-gravity in a three-dimensional, conformally reduced setting. Building on
the exact heat kernel for maximally symmetric spaces, we obtain a partial
differential equation which captures the scale-dependence of $f(R)$ for
positive and, for the first time, negative scalar curvature. The effects of
different background topologies are studied in detail and it is shown that they
affect the gravitational RG flow in a way that is not visible in
finite-dimensional truncations. Thus, while featuring local background
independence, the functional renormalization group equation is sensitive to the
topological properties of the background. The detailed analytical and numerical
analysis of the partial differential equation reveals two globally well-defined
fixed functionals with at most a finite number of relevant deformations. Their
properties are remarkably similar to two of the fixed points identified within
the $R^2$-truncation of full Quantum Einstein Gravity. As a byproduct, we
obtain a nice illustration of how the functional renormalization group realizes
the "integrating out" of fluctuation modes on the three-sphere.
