  We investigate a class of periodically driven many-body systems that allows
us to extend the phenomenon of prethermalization to the vicinity of isolated
intermediate-to-low drive frequencies away from the high-frequency limit. We
provide numerical evidence for the formation of a parametrically long-lived
prethermal plateau, captured by an effective Floquet Hamiltonian computed using
the replica inverse-frequency expansion, and demonstrate its stability
w.r.t.~random perturbations in the drive period. Considering exclusively
nonintegrable Floquet Hamiltonians, we find that heating rates are
non-universal: we observe Fermi's Golden Rule scaling, power-law scaling
inconsistent with the Golden Rule, and non-power law scaling, depending on the
drive. Despite the asymptotic character of the inverse-frequency expansion, we
show that it describes the thermostatic properties of the state all along the
evolution up to infinite temperature, with higher-order terms improving the
accuracy. Our results suggest a dynamical mechanism to gradually increase
temperature in isolated quantum simulators, such as ultracold atoms, and open
up an alternative possibility to investigate thermal phase transitions and the
interplay between thermal and quantum criticality using Floquet drives.
