Heat, by definition, is energy that comes with associated entropy. It is a well verified physical law that entropy cannot be destroyed, it can be created but the only way to decrease your amount of entropy is to move it somewhere else. To avoid having entropy build up on your spacecraft until something breaks, you need to get rid of it by moving it off the craft.
On a spacecraft, the only way you can move energy and entropy off the spacecraft is by putting it in something else and ejecting that something else, or radiating it away as electromagnetic waves. Since trying to put all your entropy in matter and dumping it is wasteful for your mass budget, radiation is the usual trick. However, the required radiator area scales as 1/temperature to the fourth power. The colder you are storing your entropy, the larger the radiator you need to get rid of it, and the size increases very rapidly with decreasing temperature.
"Maybe A Future Scientific Breakthrough Will Let Me Have My Way"
First off, from the standpoint of probability, there is at least a 50% chance that any new scientific breakthrough will actually make it harder to do what you want.
The general rule is what physicists call the correspondence principle or the Classical limit. This states that any new theory must give the same answers as the old theory where the old theory has been confirmed by experiment. Which means if you just state that in the year 2525 Professor XYZ came up with the "Take THAT, Einstein!" theory of FTL travel, you still have a problem. You have to explain how the TTE theory allows FTL flight while still giving the same answers that relativity theory did for all those experiments it confirmed. Experiments that were accurate to quite a few decimal points.
"A scientific explanation is one that is vindicated by practice." Radio transmitters transmit, and radio receivers receive. Lasers lase. Nuclear reactors react. Semi-conductors occasionally conduct. Tunnel diodes, LED's, SQUIDS, and other electromagnetic devices based on quantum mechanics do their thing repeatedly and reliably. So we're obviously doing something right! And we don't dare throw away the theoretical base on which these gadgets do indeed work. We can and should modify the theoretical base as necessary, but we can't throw it away. Any new theories of the universe must be compatible with the old ones or at least permit logical and rational modifications in order to shoe-horn the old theories into the new ones.
There is no unexplained phenomenon that might result in violating thermodynamics - and if there WERE something that violated thermo, it would radically change the universe as we know it - for instance, stellar processes require thermodynamics, the entire model of cosmology is based off of known properties for thermodynamics. Your car runs on thermodynamic processes. And all of these things work out the same way, and derive from the same knowledge base.