Fresh pineapple doesn't store long at room temperature—two days, tops. You might get an additional three days if you refrigerate it. The fact is, you can, and the process is pretty quick and simple. The trickiest part of freezing pineapple is picking out a ripe one. The peak season for fresh pineapple runs from March to July, but pineapples are available for sale in most grocery stores year-round. To choose a good pineapple, start with smell.