Ive had arguments with many a technical boffin over this and the school of thought is definately split into 3:

1) DEFFO needs fuel to cool.
2) DEFFO doesnt need fuel to cool.
3) Who Cares?

I always pose the same question though, and to me its "Virtually" this simple...

"What is coolest?"
1) An engine passing "unburnt" air through its cylinders with no fuel added to make it flammable.

2) A burning fuel mixture of any strength whatsoever.

Ive heard some interesting arguments over the years though for sure. Some of which certainly had me racking my brains, which is always nice...