Axion-like particles (ALPs) are a compelling candidate for dark matter (DM). Their production is associated with the formation of a string-wall network. This system must annihilate, producing gravitational waves and non-relativistic ALPs. In my talk I will show that these gravitational waves, if produced at temperatures below 100 eV, could be detected by future cosmological probes for ALPs with mass from 10⁻¹⁶ to 10⁶ eV, opening a window on ALP DM with arbitrary small couplings to the Standard Model.