We present our initial results on the CO rotational spectral line energy distribution (SLED) of the J to J-1 transitions from J = 4 up to 13 from Herschel SPIRE spectroscopic observations of 65 luminous infrared galaxies (LIRGs) in the Great Observatories All-Sky LIRG Survey. The observed SLEDs change on average from one peaking at J < or =, slant 4 to a broad distribution peaking around J ~ 6 to 7 as the IRAS 60-to-100 mu m color, C(60/100), increases. However, the ratios of a CO line luminosity to the total infrared luminosity, L sub(IR), show the smallest variation for J around 6 or 7. This suggests that, for most LIRGs, ongoing star formation (SF) is also responsible for a warm gas component that emits CO lines primarily in the mid-J regime (5 <, ~ J <, ~ 10). As a result, the logarithmic ratios of the CO line luminosity summed over CO (5-4), (6-5), (7-6), (8-7) and (10-9) transitions to L sub(IR), log R sub(midCO), remain largely independent of C(60/100), and show a mean value of -4.13 (= log (ProQuest: Formulae and/or non-USASCII text omitted)) and a sample standard deviation of only 0.10 for the SF-dominated galaxies. Including additional galaxies from the literature, we show, albeit with a small number of cases, the possibility that galaxies, which bear powerful interstellar shocks unrelated to the current SF, and galaxies, in which an energetic active galactic nucleus contributes significantly to the bolometric luminosity, have their R sub(midCO) higher and lower than (ProQuest: Formulae and/or non-USASCII text omitted) respectively.