No.
There is no circulation of hydraulic fluid if the brakes are pressurised, the cooling issue is to do with whether the brakes are ON or not, as expounded earlier in this thread.
My original response reflected my concern that one day, perhaps soon, we'll see a crew come to a stop after an RTO, then not set the parking brake, then do something (spot that the autobrake switch is at RTO, and switch it off, or de-power the aircraft), for the aircraft to roll unbraked into the next obstacle downhill and/or downwind of it. Thus, once again, a trained flight crew (Boeing's words, not mine) will snatch defeat from the jaws of victory, and probably end up on the front page of a tabloid and/or in the fleet manager's office (perhaps simultaneously).
This response reflects a concern that it's now possible to be an instructor or examiner with very little technical knowledge of the aircraft's systems - or system design in general.
