The barocaloric effect of a solid material is an intense research topic due to its potential application in solid-state refrigeration. Among the proposed candidates, elastic polymers are distinctive because their barocaloric responses are independent from a pressure-induced phase transition which makes it possible to realize a broad working temperature range in principle. However, the barocaloric performance of most elastic polymers diminishes significantly as temperature decreases. In this work, giant and reversible barocaloric effects were observed in a broad working temperature range from 252 to 345 K in an amorphous polymer of ethylene propylene diene monomer, which are much higher than the investigated crystalline and partially crystallized ones. It is demonstrated that the degree of crystallinity can be a key factor responsible for the mobility of polymer chains and the corresponding barocaloric performance at low temperatures. The reversible giant barocaloric effects, broad working temperature regions, low cost, and absence of pressure-transmitting fluid make the ethylene propylene diene monomer attractive for solid-state barocaloric refrigeration.