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}.
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