  The epoch when the Universe had a temperature higher than a GeV is long
before any time at which we have reliable observations constraining the
cosmological evolution. For example, the occurrence of a second burst of
inflation (sometimes called thermal inflation) at a lower energy scale than
standard inflation, or a short epoch of early matter domination, cannot be
ruled out by present cosmological data. The cosmological stochastic
gravitational wave background, on scales accessible to interferometer
detection, is sensitive to non-standard cosmologies of this type. We consider
the implications of such alternative models both for ground-based experiments
such as LIGO and space-based proposals such as LISA. We show that a second
burst of inflation leads to a scale-dependent reduction in the spectrum.
Applied to conventional inflation, this further reduces an already
disappointingly low signal. In the pre big bang scenario, where a much more
potent signal is possible, the amplitude is reduced but the background remains
observable by LISA in certain parameter space regions. In each case, a second
epoch of inflation induces oscillatory features into the spectrum in a manner
analogous to the acoustic peaks in the density perturbation spectrum. On LIGO
scales, perturbations can only survive through thermal inflation with
detectable amplitudes if their amplitudes were at one time so large that linear
perturbation theory is inadequate. Although for an epoch of early matter
domination the reduction in the expected signal is not as large as the one
caused by a second burst of inflation, the detection in the context of the pre
big bang scenario may not be possible since the spectrum peaks around the LIGO
frequency window and for lower frequencies behaves as $f^3$.
