The heavy rare-earth-based Laves phases are well-studied intermetallic materials that stand out for their remarkably high magnetocaloric effects, particularly at cryogenic temperatures. In this study, we present the findings of our comprehensive investigation of cobalt Laves phases RCo₂ with R standing for erbium, holmium, dysprosium, and terbium. This includes the determination of the magnetocaloric effect by indirect methods using calorimetric and magnetization data. Furthermore, for the first time in these materials, we directly measured the adiabatic temperature change at high magnetic fields up to 20 T. The largest ΔT_ad value of 17 K, we obtained for ErCo₂. Because the order of the transition significantly impacts the efficiency of thermodynamic cycles, we have also focused on determining the transition order in these materials. This was done through the application of established methods and a recently proposed quantitative criterion including the value of the local exponent n. Further, we compare our results with other materials using a straightforward material-based figure of merit - the temperature-averaged entropy change (TEC). Our results demonstrate the great potential of these materials for applications such as for magnetic hydrogen liquefaction.