  Efforts to understand the origin and growth of massive black holes observed
in the early Universe have spurred a strong interest in the evolution and fate
of rapidly-accreting primordial (metal-free) stars. Here, we investigate the
evolution of such Population III stars under variable accretion rates, focusing
on the thermal response and stellar structure, the impact of the luminosity
wave encountered early in the pre-main sequence phase, and the influence of
accretion on their subsequent evolution. We employ the Geneva stellar evolution
code and simulate ten models with varying accretion histories, covering a final
mass range from 491 M$_{\odot}$ to 6127 M$_{\odot}$. Our findings indicate that
the critical accretion rate delineating the red and blue supergiant regimes
during the pre-main sequence evolution is approximately $2.5\times10^{-2}
$M$_{\odot}$/yr. Once core hydrogen burning commences, the value of this
critical accretion rate drops to $7.0\times10^{-3}$M$_{\odot}$/yr. Moreover, we
also confirm that the Kelvin-Helmholtz timescale in the outer surface layers is
the more relevant timescale for determining the transition between red and blue
phases. Regarding the luminosity wave, we find that it affects only the early
pre-main sequence phase of evolution and does not directly influence the
transition between red and blue phases, which primarily depends on the
accretion rate. Finally, we demonstrate that variable accretion rates
significantly impact the lifetimes, surface enrichment, final mass and time
spent in the red phase. Our study provides a comprehensive understanding of the
intricate evolutionary patterns of Population III stars subjected to variable
accretion rates.
