We present far-infrared (57-196 μm) spectra of 21 protostars in the Orion molecular clouds. These were obtained with the Photodetector Array Camera and Spectrometer (PACS) on board the Herschel Space observatory as part of the Herschel Orion Protostar Survey program. We analyzed the emission lines from rotational transitions of CO, involving rotational quantum numbers in the range J_(up) = 14-46, using PACS spectra extracted within a projected distance of ≾2000 AU centered on the protostar. The total luminosity of the CO lines observed with PACS (L_(CO)) is found to increase with increasing protostellar luminosity (L_(bol)). However, no significant correlation is found between L_(CO) and evolutionary indicators or envelope properties of the protostars such as bolometric temperature, T_(bol), or envelope density. The CO rotational (excitation) temperature implied by the line ratios increases with increasing rotational quantum number J, and at least 3–4 rotational temperature components are required to fit the observed rotational diagram in the PACS wavelength range. The rotational temperature components are remarkably invariant between protostars and show no dependence on L_(bol), T_(bol), or envelope density, implying that if the emitting gas is in local thermodynamic equilibrium, the CO emission must arise in multiple temperature components that remain independent of L_(bol) over two orders of magnitudes. The observed CO emission can also be modeled as arising from a single-temperature gas component or from a medium with a power-law temperature distribution; both of these require sub-thermally excited molecular gas at low densities (n(H_2) ≾ 10^6 cm^(–3)) and high temperatures (T≳2000 K). Our results suggest that the contribution from photodissociation regions, produced along the envelope cavity walls from UV-heating, is unlikely to be the dominant component of the CO emission observed with PACS. Instead, the "universality" of the rotational temperatures and the observed correlation between L_(CO) and L_(bol) can most easily be explained if the observed CO emission originates in shock-heated, hot (T≳2000 K), sub-thermally excited (n(H_2) ≾ 10^6 cm^(–3)) molecular gas. Post-shock gas at these densities is more likely to be found within the outflow cavities along the molecular outflow or along the cavity walls at radii ≳ several 100-1000 AU.