  The formation of SMSs is a potential pathway to seed SMBHs in the early
universe. A critical issue for forming SMSs is stellar UV feedback, which may
limit the stellar mass growth via accretion. In this paper we study the
evolution of an accreting SMS and its UV emissivity under conditions of
realistic variable accretion from a self-gravitating circumstellar disc. First
we conduct a 2D hydrodynamical simulation to follow the long-term protostellar
accretion until the stellar mass exceeds $10^4~M_\odot$. The disc fragments due
to gravitational instability, creating a number of small clumps that rapidly
migrate inward to fall onto the star. The resulting accretion history is thus
highly time-dependent: short episodic accretion bursts are followed by longer,
relative quiescent phases. We show that the circumstellar disc for the
so-called direct collapse model is more unstable and generates greater
variability over shorter timescales than normal Pop III cases. We conduct a
post-process stellar evolution calculation using the obtained accretion
history. Our results show that, regardless of the strong variability of the
accretion rates, the stellar radius monotonically increases with almost
constant effective temperature at $T_{\rm eff} \simeq 5000$ K as the stellar
mass increases. The resulting UV feedback is too weak to hinder mass accretion
due to the low flux of stellar UV photons, thus verifying our implicit
assumption of no stellar feedback during the hydrodynamic simulations. The
insensitivity of stellar evolution to variable accretion is attributed to the
fact that typical timescales of variability, $\lesssim 10^3$ years, are too
short to affect the stellar structure. We argue that this evolution will
continue until the SMS eventually collapses to produce a massive black hole by
the general relativistic instability after the stellar mass reaches $\gtrsim
10^5~M_\odot$.
