Metal–organic frameworks (MOFs) that display step-shaped adsorption isotherms, i.e., “phase-change” MOFs, represent a relatively small subset of all known MOFs. Yet, they are rapidly emerging as promising sorbents to achieve excellent gas separation performances with little energy demand. In this work, we assessed F4_MIL-140A(Ce), a recently discovered “phase-change” MOF adsorbent, for CO2 capture in two scenarios using a pressure-vacuum swing adsorption process, namely a coal-fired power plant flue gas (12.5%mol CO2), and a steel plant flue gas (25.5%mol CO2). Four CO2 and three N2 adsorption isotherms were collected on F4_MIL-140A(Ce) over a range of temperatures and modelled using a bespoke equation for step-shaped isotherms. We accurately measured the heat capacity of F4_MIL-140A(Ce), a key thermodynamic property for a sorbent, using a method based on differential scanning calorimetry that overcomes the issues associated with the poor thermal conductivity of MOF powders. We then used these experimental data as input in a process optimisation framework and we compared the CO2 capture performance of F4_MIL-140A(Ce) to that of other “canonical” sorbents, including, zeolite 13X, activated carbon and three MOFs (i.e., HKUST-1, UTSA-16 and CALF-20). We found that F4_MIL-140A(Ce) has the potential to perform better than other sorbents, in terms of recovery and purity, under most of the simulated process conditions. We attribute such promising performance to the non-hysteretic step-shaped isotherm, the low uptake capacity for N2 and the mild heat of CO2 adsorption displayed by F4_MIL-140A(Ce).