Ideally, a peach cobbler is served warm. You need the cobbler to cool a little bit so the juices will thicken back up. You can store a peach cobbler for a day or two on the counter, covered, sitting at room temperature. But, it iss always safer to store it in the refrigerator. It should last covered in the fridge for up to one week. You can reheat small portions of peach cobbler in the microwave if you enjoy it warm.