  This paper demonstrates cooling of the center-of-mass motion of 10
$\mu$m-diameter optically levitated silica spheres to an effective temperature
of $50\pm22 \mu$K, achieved by minimizing the technical pointing noise of the
trapping laser. This low noise leads to an acceleration and force sensitivity
of $95\pm41$ n$g/\sqrt{\mathrm{Hz}}$ ($g = 9.8$ m/s$^2$) and $0.95\pm0.11$
aN$/\sqrt{\mathrm{Hz}}$, respectively, at frequencies near 50 Hz. This force
sensitivity is comparable to that demonstrated for optically levitated
nanospheres that are $10^4$ times less massive, corresponding to an
acceleration sensitivity that is several orders of magnitude better. It is
further shown that under these conditions the spheres remain stably trapped at
pressures of $\sim 10^{-7}$ mbar with no active cooling for periods longer than
a day. Feedback cooling is still necessary in the moderate-pressure regime,
motivating a comprehensive study of the loss mechanisms of the microspheres and
providing better understanding of the requirements for feedback-free optical
trapping in vacuum. This work can enable high-sensitivity searches for
accelerations and forces acting on micron-sized masses, including those that
could be produced by new physics beyond the Standard Model.
