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Particle physics beyond the Standard Model can temporarily induce an equation of state for the early Universe which is not that of radiation. The resulting non-standard expansion histories leave signatures in primordial stochastic backgrounds of gravitational waves (GW). This talk focuses on how early dynamics of the axion can induce a matter-kination era inside the standard radiation era. The matter-kination era imprints a smoking-gun GW peak, whose position depends on the energy scale and duration of the kination era. Remarkably, the GW peak lies inside LISA, Einstein Telescope, and Cosmic Explorer sensitivity windows. Future GW observatories will thus offer a new window on axion models.