Three metal-organic frameworks with the general formula Co(BPZX) (BPZX2- = 3-X-4,4'-bipyrazolate, X = H, NH2, NO2) constructed with ligands having different functional groups on the same skeleton have been employed as heterogeneous catalysts for aerobic liquid-phase oxidation of cumene with O2 as oxidant. O2 adsorption isotherms collected at pO2 = 1 atm and T = 195 and 273 K have cast light on the relative affinity of these catalysts for dioxygen. The highest gas uptake at 195 K is found for Co(BPZ) (3.2 mmol/g (10.1 wt % O2)), in line with its highest BET specific surface area (926 m2/g) in comparison with those of Co(BPZNH2) (317 m2/g) and Co(BPZNO2) (645 m2/g). The O2 isosteric heat of adsorption (Qst) trend follows the order Co(BPZ) > Co(BPZNH2) > Co(BPZNO2). Interestingly, the selectivity in the cumene oxidation products was found to be dependent on the tag present in the catalyst linker: while cumene hydroperoxide (CHP) is the main product obtained with Co(BPZ) (84% selectivity to CHP after 7 h, pO2 = 4 bar, and T = 363 K), further oxidation to 2-phenyl-2-propanol (PP) is observed in the presence of Co(BPZNH2) as the catalyst (69% selectivity to PP under the same experimental conditions).