We study the Ricci flow of initial metrics which are C⁰-perturbations of the hyperbolic metric on Hⁿ. If the perturbation is bounded in the L²-sense, and small enough in the C⁰-sense, then we show the following: In dimensions four and higher, the scaled Ricci harmonic map heat flow of such a metric converges smoothly, uniformly and exponentially fast in all C^k-norms and in the L²-norm to the hyperbolic metric as time approaches infinity. We also prove a related result for the Ricci flow and for the two-dimensional conformal Ricci flow.