  Reionization leads to large spatial fluctuations in the intergalactic
temperature that can persist well after its completion. We study the imprints
of such fluctuations on the $z\sim5$ Ly$\alpha$ forest flux power spectrum
using a set of radiation-hydrodynamic simulations that model different
reionization scenarios. We find that large-scale coherent temperature
fluctuations bring $\sim20-60\%$ extra power at $k\sim0.002$ s/km, with the
largest enhancements in the models where reionization is extended or ends the
latest. On smaller scales ($k\gtrsim0.1$ s/km), we find that temperature
fluctuations suppress power by $\lesssim10\%$. We find that the shape of the
power spectrum is mostly sensitive to the reionization midpoint rather than
temperature fluctuations from reionization's patchiness. However, for all of
our models with reionization midpoints of $z\le 8$ ($z\le 12$) the shape
differences are $\lesssim20\%$ ($\lesssim40\%$) because of a surprisingly
well-matched cancellation between thermal broadening and pressure smoothing
that occurs for realistic thermal histories. We also consider fluctuations in
the ultraviolet background, finding their impact on the power spectrum to be
much smaller than temperature fluctuations at $k\gtrsim0.01$ s/km. Furthermore,
we compare our models to power spectrum measurements, finding that none of our
models with reionization midpoints of $z<8$ is strongly preferred over another
and that all of our models with midpoints of $z\geq8$ are excluded at
$2.5\sigma$. Future measurements may be able to distinguish between viable
reionization models if they can be performed at lower $k$ or, alternatively, if
the error bars on the high-$k$ power can be reduced by a factor of $1.5$.
