  The highly populated, low-energy excitations of a scalar field of mass
$m_a\sim 10^{-22}\,\textrm{eV}$ can represent the full dark matter content of
the universe and alleviate some tensions in the standard cosmological scenario
on small-scales. This {\it fuzzy dark matter} component is commonly assumed to
arise as the consequence of a new axion-like particle in the matter sector, yet
for simplicity it is usually modeled in terms of a simple free quadratic field.
In this paper we consider how the cosmological constraints are modified when
the effects of an instanton potential and temperature-dependent mass are
included. Current isocurvature and tensor bounds confirms that this particle
should be formed before the end of a low-scale inflation period with Hubble
parameter $H_I\lesssim 2.5\times 10^{12}\,\textrm{GeV}$, in accordance with
previous free-field analysis. The axion decay constant, $f_a$, which fixes
axion couplings, appears in the instanton potential and determines the relic
abundance, the stability of galaxy cores to axion emission, and the direct
searches of fuzzy dark matter. If the axion mass is $T$-independent, we find
that $f_a\gtrsim 10^{16}\,\text{GeV}$ is required in order to reach the
observed relic density without fine tuning the initial conditions, while for a
$T$-dependent case this bound can be lowered by an order of magnitude. However,
the anharmonicities in the instanton potential, and mainly a $T$-dependent
mass, can delay the onset of field oscillations, leading to larger physical
suppression scales in the matter power spectrum for a fixed zero-temperature
axion mass. This may favor a string or accidental axion over one emerging from
a strongly coupled gauge sector if this model is required to provide large
galactic halo cores while simultaneously satisfying observational constraints
from cosmic structure formation.
