We present a comparison of the molecular gas properties in the outflow vs. in the ambient medium of the local prototype radio-loud and ultraluminous-infrared galaxy 4C12.50 (IRAS 13451+1232), using new data from the IRAM Plateau de Bure Interferometer and 30 m telescope and from the Herschel space telescope. Previous  H2 (0–0) S(1) and S(2) observations with the Spitzer space telescope had indicated that the warm (~400 K) molecular gas in 4C12.50 is made up of a 1.4( ± 0.2) × 108M⊙ ambient reservoir and a 5.2(±1.7) × 107M⊙ outflow. The new CO(1–0) data cube indicates that the corresponding cold (25 K)  H2 gas mass is 1.0(±0.1) × 1010M⊙ for the ambient medium and < 1.3 × 108 M⊙ for the outflow, when using a CO-intensity-to-H2-mass conversion factor α of 0.8 M⊙/(K km s-1 pc2). The combined mass outflow rate is high, 230–800 M⊙/yr, but the amount of gas that could escape the galaxy is low. A potential inflow of gas from a 3.3(±0.3) × 108M⊙ tidal tail could moderate any mass loss. The mass ratio of warm-to-cold molecular gas is ≳30 times higher in the outflow than in the ambient medium, indicating that a non-negligible fraction of the accelerated gas is heated to temperatures at which star formation is inefficient. This conclusion is robust against the use of different α factor values and/or different warm gas tracers (H2 vs.  H2 plus CO). With the CO-probed gas mass at least 40 times lower at 400 K than at 25 K, the total warm-to-cold mass ratio is always lower in the ambient gas than in the entrained gas. Heating of the molecular gas could facilitate the detection of new outflows in distant galaxies by enhancing their emission in intermediate rotational number CO lines.