  Resonance modes in single crystal sapphire ($\alpha$-Al$_2$O$_3$) exhibit
extremely high electrical and mechanical Q-factors ($\approx 10^9$ at 4K),
which are important characteristics for electromechanical experiments at the
quantum limit. We report the first cooldown of a bulk sapphire sample below
superfluid liquid helium temperature (1.6K) to as low as 25mK. The
electromagnetic properties were characterised at microwave frequencies, and we
report the first observation of electromagnetically induced thermal bistability
in whispering gallery modes due to the material $T^3$ dependence on thermal
conductivity and the ultra-low dielectric loss tangent. We identify "magic
temperatures" between 80 to 2100 mK, the lowest ever measured, at which the
onset of bistability is suppressed and the frequency-temperature dependence is
annulled. These phenomena at low temperatures make sapphire suitable for
quantum metrology and ultra-stable clock applications, including the possible
realization of the first quantum limited sapphire clock.
