  Rotations of axion fields in the early universe can produce dark matter and
the matter-antimatter asymmetry of the universe. We point out that the rotation
can generate an observable amount of a stochastic gravitational-wave (GW)
background. It can be doubly enhanced in a class of models in which the
equation of state of the rotations rapidly changes from a non-relativistic
matter-like one to a kination-like one by 1) the so-called poltergeist
mechanism and 2) slower redshift of GWs compared to the axion kination fluid.
In supersymmetric UV completion, future GW observations can probe the
supersymmetry-breaking scale up to $10^7\,$GeV even if the axion does not
directly couple to the Standard Model fields.
