Assuming that inflation is followed by a phase where the energy density of the Universe is dominated by a component with a general equation of state, we evaluate the spectrum of primordial gravitational waves induced in the postinflationary Universe. We show that if the energy density of the Universe is dominated by a component $\ensuremath{\phi}$ before big bang nucleosynthesis, its equation of state could be constrained by gravitational wave experiments depending on the ratio of energy densities of $\ensuremath{\phi}$ and radiation and also the temperature at the end of the $\ensuremath{\phi}$-dominated era. Also, we discuss the impact of scale dependence of tensor modes on the primordial gravitational wave spectrum during the $\ensuremath{\phi}$ domination. These models are motivated by beyond Standard Model physics and scenarios for nonthermal production of dark matter in the early Universe. We also constrain the parameter space of the tensor spectral index and the tensor-to-scalar ratio, using the experimental limits from gravitational wave experiments.