  We report an investigation of laser frequency stabilization using a
whispering gallery mode resonator that is temperature stabilized by a dual-mode
technique. This dual-mode technique has yielded mode volume temperature
instabilities at the nK level, suggesting that high frequency stability may
also be reached. Here, we experimentally and theoretically investigate the
dynamics of such a system and the important factors affecting the achievable
frequency stability. We calculate that the dual-mode technique can reduce the
effective fractional temperature coefficient of the reference system to
3.6x10^-8 K^-1 within the temperature feedback bandwidth. We demonstrate a 1560
nm laser stabilized to 1.3x10^-12 at 1 s and 1.1x10^-10 at 1000 s,
corresponding to a long-term drift of 21 kHz/hr.
