Do any aircraft have liquid-cooled brakes?
A liquid heat exchanger system for brake discs, with coolant, lines, some way of transferring heat from the disc pack to the heat exchanger, that would somehow keep the entire apparatus liquid tight... yikes.  I can only imagine what a monstrosity such a design would be, even if it's technically feasible.

Brake temperature requirements don't affect gate turnaround times. 99.99% of the time you land and taxi in, and an hour later, off you go, long after they've cooled off.  On a reject, on the rare occasion the reject didn't require a return to gate for maintenance action of some kind, you have the Brake Temperature Monitoring System to tell you when the brakes are good to go even if you don't return to gate (most rejects are because of a Caution or Warning prior to V1 -  and if you reject in the "high energy" phase, usually above 80 or 90 kt, which is a callout during the roll, your airline will likely require you to return for a brake and tire inspection anyway, "liquid cooled" or not).  If the BTMS is MEL'ed, you will have to observe a minimum brake cool down time as per the associated MMEL Ops Procedure.

So yes there are times when regular brakes require you to sit and wait for a period of time, but it would happen on 1 departure in, well, thousands (departure delays from brake overheating, from pilots riding the brakes during taxi, is probably a bigger problem operationally).   In any case, in view of the actual operational impact, nobody would want to pay for such a cooling system even if it was feasible.