  The scalar perturbation induced gravitational waves are a probe of the
primordial density perturbation spectrum on small scales. In this paper, we
show that they can also probe the thermal history of the universe. We assume
the universe underwent a stage with a constant equation of state parameter $w$,
followed by the radiation-dominated stage of the conventional big bang
universe. We find that the infrared slope of the power spectrum of the induced
stochastic gravitational wave background for decelerating cosmologies is
related to the equation of state of the universe. Furthermore, the induced
gravitational wave spectrum has in general a broken power-law shape around the
scale of reheating. Interestingly, below the threshold $w=0$ of the equation of
state parameter, the broken power-law presents a peak for a Dirac delta peak in
the scalar spectrum. For a finite width peak, the threshold changes to
$w=-1/15$ depending on the value of the width. In some cases, such a broken
power-law gravitational wave spectrum may degenerate to the spectrum from other
sources like phase transitions or global cosmic strings.
