  The cryogenic sapphire oscillator (CSO) at the Paris Observatory has been
continuously compared to various Hydrogen Masers since 2001. The early data
sets were used to test Local Lorentz Invariance in the Robertson-Mansouri-Sexl
(RMS) framework by searching for sidereal modulations with respect to the
Cosmic Microwave Background, and represent the best Kennedy-Thorndike
experiment to date. In this work we present continuous operation over a period
of greater than six years from September 2002 to December 2008 and present a
more precise way to analyze the data by searching the time derivative of the
comparison frequency. Due to the long-term operation we are able to search both
sidereal and annual modulations. The results gives P_{KT} =
\beta_{RMS}-\alpha_{RMS}-1 = -1.7(4.0) \times 10^{-8} for the sidereal and
-23(10) \times 10^{-8} for the annual term, with a weighted mean of -4.8(3.7)
\times 10^{-8}, a factor of 8 better than previous. Also, we analyze the data
with respect to a change in gravitational potential for both diurnal and annual
variations. The result gives \beta_{H-Maser} - \beta_{CSO} = -2.7(1.4) \times
10^{-4} for the annual and -6.9(4.0) \times 10^{-4} for the diurnal terms, with
a weighted mean of -3.2(1.3) \times 10^{-4}. This result is two orders of
magnitude better than other tests that use electromagnetic resonators. With
respect to fundamental constants a limit can be provided on the variation with
ambient gravitational potential and boost of a combination of the fine
structure constant (\alpha), the normalized quark mass (m_q), and the electron
to proton mass ratio (m_e/m_p), setting the first limit on boost dependence of
order 10^{-10}.
