  We investigate the effects of an early cosmological period dominated by
primordial 2-2-holes on axion dark matter. The 2-2-holes emerge as a new family
of horizonless classical solutions for ultracompact matter distributions in
quadratic gravity, a candidate theory of quantum gravity. Thermal 2-2-holes,
sourced by relativistic thermal gas, exhibit Hawking-like radiation and fulfill
the entropy-area law before they become remnants with almost no radiation. In
this paper, we consider the remnant contribution to dark matter (DM) small and
adopt the axion DM scenario by the misalignment mechanism. We show that a
2-2-hole domination phase in the evolution of the universe changes the axion
mass window obtained from the dark matter abundance constraints. The biggest
effect occurs when the remnants have the Planck mass, which is the case for a
strongly coupled quantum gravity. The change in abundance constraints for the
Planck mass 2-2-hole remnants amounts to that of the primordial black hole
(PBH) counterpart. Therefore, since we use the revised constraints from
gravitational waves on the initial fraction of 2-2-holes, the results here can
also be considered the updated version of the PBH case. As a result, the lower
limit on the axion mass is found as $m_a \sim10^{-9}$ eV. Furthermore, the
domination scenario itself constrains the remnant mass $ M_{\mathrm{min}}$
considerably. Given that we focus on the pre-BBN domination scenario in order
not to interfere with BBN (Big Bang nucleosynthesis) constraints, the remnant
mass window becomes $m_{\mathrm{Pl}} \lesssim M_{\mathrm{min}} \lesssim
0.1\;\rm{g}$.
