We investigate the physical conditions of the CO gas, based on the submillimeter imaging spectroscopy from a 2′ × 7′ (1.5 × 5 pc²) area near the young star cluster, Trumpler 14 of the Carina Nebula. The observations presented in this work are taken with the Fourier Transform Spectrometer (FTS) of the Spectral and Photometric Imaging REceiver (SPIRE) onboard the Herschel Space Observatory. The newly observed spectral lines include [CI] 370 μm [CI] 609 μm, and CO transitions from J = 4−3 to J = 13−12. Our field of view covers the edge of a cavity carved by Trumpler 14 about 1 Myr ago and marks the transition from H ii regions to photo-dissociation regions. The observed CO intensities are the most prominent at the northwest region, Car I-E. With the state-of-the-art Meudon PDR code, we successfully derive the physical conditions, which include the thermal pressure (P) and the scaling factor of radiation fields (G_UV), from the observed CO spectral line energy distributions (SLEDs) in the observed region. The derived G_UV values generally show excellent agreement with the UV radiation fields created by nearby OB-stars and thus confirm that the main excitation source of the observed CO emission is the UV-photons provided by the massive stars. The derived thermal pressure is in the range 0.5−3 × 10⁸ K cm^-3 with the highest values found along the ionization front in Car I-E region facing Trumpler 14, hinting that the cloud structure is similar to the recent observations of the Orion Bar. We also note a discrepancy at a local position (<0.17 × 0.17 pc²) between the photo-dissociation region (PDR) modeling result and the UV radiation fields estimated from nearby massive stars, which requires further investigation on nearby objects that could contribute to local heating, including outflow. Comparing the derived thermal pressure with the radiation fields, we report the first observationally derived and spatially resolved P ~ 2 × 10⁴ G_UV relationship. As direct comparisons of the modeling results to the observed ¹³CO, [O I] 63 μm, and [C II] 158 μm intensities are not straightforward, we urge the reader to be cautious when constraining the physical conditions of PDRs with combinations of ¹²CO, ¹³CO, [C I], [O I] 63 μm, and [C II] 158 μm observations.