  The primordial irreducible gravitational-wave background due to quantum
vacuum tensor fluctuations produced during inflation spans a large range of
frequencies with an almost scale-invariant spectrum but is too low to be
detected by the next generation of gravitational-wave interferometers. We show
how this signal is enhanced by a short temporary kination era in the
cosmological history (less than 10 e-folds), that can arise at any energy scale
between a GeV and the inflationary scale $10^{16}$ GeV. We argue that such
kination era is naturally generated by a spinning axion before it gets trapped
by its potential. It is usually assumed that the axion starts oscillating
around its minimum from its initially frozen position. However, the early
dynamics of the Peccei-Quinn field can induce a large kinetic energy in the
axion field, triggering a kination era, either before or after the axion
acquires its mass, leading to a characteristic peak in the primordial
gravitational-wave background. This represents a smoking-gun signature of axion
physics as no other scalar field dynamics is expected to trigger such a
sequence of equations of state in the early universe. We derive the resulting
gravitational-wave spectrum, and present the parameter space that leads to such
a signal as well as the detectability prospects, in particular at LISA,
Einstein Telescope, Cosmic Explorer and Big Bang Observer. We show both
model-independent predictions and present as well results for two specific
well-motivated UV completions for the QCD axion dark matter where this dynamics
is built-in.
