  We use a functional renormalization group equation tailored to the
Arnowitt-Deser-Misner formulation of gravity to study the scale-dependence of
Newton's coupling and the cosmological constant on a background spacetime with
topology S^1xS^d. The resulting beta functions possess a non-trivial
renormalization group fixed point, which may provide the high-energy completion
of the theory through the asymptotic safety mechanism. The fixed point is
robust with respect to changing the parametrization of the metric fluctuations
and regulator scheme. The phase diagrams show that this fixed point is
connected to a classical regime through a crossover. In addition the flow may
exhibit a regime of "gravitational instability", modifying the theory in the
deep infrared. Our work complements earlier studies of the gravitational
renormalization group flow on a background topology S^1xT^d and establishes
that the flow is essentially independent of the background topology.
