  We study pseudo-Goldstone dark matter in the $\mathbb{Z}_{3}$ complex scalar
singlet model. Because the direct detection spin-independent cross section is
suppressed, such dark matter is allowed in a large mass range. Unlike in the
original model stabilized by a parity, due to the cubic coupling of the singlet
the $\mathbb{Z}_{3}$ model can accommodate first-order phase transitions that
give rise to a stochastic gravitational wave signal potentially observable in
future space-based detectors.
