  Gravitons radiated from light, evaporating black holes contribute to the
stochastic background of gravitational waves. The spectrum of such emission
depends on both the mass and the spin of the black holes, as well as on the
redshifting that occurs between the black hole formation and today. This, in
turn, depends on the expansion history of the universe, which is largely
unknown and unconstrained at times prior to the synthesis of light elements.
Here, we explore the features of the stochastic background of gravitational
waves from black hole evaporation under a broad range of possible early
cosmological histories. The resulting gravitational wave signals typically peak
at very high frequencies, and offer opportunities for proposed ultra-high
frequency gravitational wave detectors. Lower-frequency peaks are also
possible, albeit with a suppressed intensity that is likely well below the
threshold of detectability. We find that the largest intensity peaks correspond
to cosmologies dominated by fluids with equations of state that have large
pressure-to-density ratios. Such scenarios can be constrained on the basis of
violation of $\Delta N_{\rm eff}$ bounds.
