We use a total energy difference approach to explore the ability of various density functional theory based methods in accounting for the differential effect of static electron correlation on the C(1s) and O(1s) core level binding energies (BEs) of the CO molecule. In particular, we focus on the magnitude of the errors of the computed C(1s) and O(1s) BEs and on their relative difference as compared to experiment and to previous results from explicitly correlated wave functions. Results show that the different exchange-correlation functionals studied here behave rather erratically and a considerable number of them lead to large errors in the BEs and/or the BE shifts. Nevertheless, the TPSS functional, its TPSSm and RevTPSS derivations, and its corresponding hybrid counterpart, TPSSh, perform better than average and provide BEs and BE shifts in good agreement with experiment.