  We derive the flow equation for the gravitational effective average action in
an $f(R)$ truncation on hyperbolic spacetimes using the exponential
parametrization of the metric. In contrast to previous works on compact spaces,
we are able to evaluate traces exactly using the optimised cutoff. This reveals
in particular that all modes can be integrated out for a finite value of the
cutoff due to a gap in the spectrum of the Laplacian, leading to the effective
action. Studying polynomial solutions, we find poorer convergence than has been
found on compact spacetimes even though at small curvature the equations only
differ in the treatment of certain modes. In the vicinity of an asymptotically
free fixed point, we find the universal beta function for the $R^2$ coupling
and compute the corresponding effective action which involves an $R^2 \log R^2$
quantum correction.
