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We present time-series near-infrared spectra for the classical Cepheid, CP Cephei, from the Astrophysical Research Consortium 3.5-m telescope and near-infrared spectrograph, TripleSpec , at Apache Point Observatory, NM, USA. Spectral observations were made at nine points through the minimum and partway up the ascending portion of the optical light curve for the star. Carbon monoxide (CO) was detected in absorption in the 2.3- |$\mu$| m region for each observation. We measured the strength of CO absorption using the 2-0 band head in the feature for each observation and confirm that the CO varies with pulsation. We show that these measurements follow the |$(J-K)$| colour curve, confirming that temperature drives the destruction of CO. By obtaining convolved filter magnitudes from the spectral data we found that the effect of the CO feature on K magnitudes is small, unlike the CO feature in the mid-infrared at 4.5 |$\mu$| m. The dissociation of CO in the near-infrared spectra tracks with the effect seen in the mid-infrared photometric measurements of a similar Galactic Cepheid. Confirmation of the varying CO feature illustrates the need for further investigations into the related mid-infrared period–colour–metallicity relation in order to address the impact of Cepheid metallicities on the Hubble tension. [ABSTRACT FROM AUTHOR]