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We study the formation of primordial black holes (PBHs) from densityfluctuations due to supercooled phase transitions (PTs) triggered in anaxion-like particle (ALP) model. We find that the mass of the PBHs is inverselycorrelated with the ALP decay constant $f_a$. For instance, for $f_a$ varyingfrom ${\cal O}$(100 MeV) to ${\cal O}$($10^{12}$ GeV), the PBH mass variesbetween $(10^{3} - 10^{-24}) M_{\odot}$. We then identify the ALP parameterspace where the PBH can account for the entire (or partial) dark matterfraction of the Universe, in a single (multi-component) dark matter scenario,with the ALP being the other dark matter candidate. The PBH parameter spaceruled out by current cosmological and microlensing observations can thus bedirectly mapped onto the ALP parameter space, thus providing new bounds onALPs, complementary to the laboratory and astrophysical ALP constraints.Similarly, depending on the ALP couplings to other Standard Model particles,the ALP constraints on $f_a$ can be translated into a lower bound on the PBHmass scale. Moreover, the supercooled PT leads to a potentially observablestochastic gravitational wave (GW) signal at future GW observatories, such asaLIGO, LISA and ET, that acts as another complementary probe of the ALPs, aswell as of the PBH dark matter. Finally, we show that the recent NANOGravsignal of stochastic GW in the nHz frequency range can be explained in ourmodel with $f_a\simeq (10~{\rm GeV}-1~{\rm TeV})$.

Slaying Axion-Like Particles via Gravitational Waves and Primordial Black Holes from Supercooled Phase Transition