According to StillTasty, cornbread does not need to be refrigerated. Typically, cornbread lasts around two days when left out. The best way to store cornbread is by covering it with plastic wrap or placing it in a heavy-duty freezer bag.
If refrigerated, cornbread typicallly lasts for up to a week. To refrigerate cornbread, place it in a freezer bag or wrap it in aluminum foil and store it away from moisture. Cornbread lasts the longest when it is frozen. The best way to freeze cornbread is to tightly wrap it in aluminum foil and place it in a freezer bag. Generally, frozen cornbread lasts up to three months.