I don’t know why you would want to be at the airport six hours before you flight departs, but maybe you have bad experiences with long-distance trains or the connection was simply so much better or something. In any case, the most obvious solution for ‘some place warm’ is simply to go into the airport itself.
Since it is the largest airport in Germany and one of the largest in Europe with flights all day round (except for the night hours) you can always just enter the building. There are shops and cafés landside. Maybe you can even check in that early, especially if your flight is with one of the major airlines (e.g. Lufthansa), which would allow you to enter the airside area where there are more shops and cafés.