For a pocket universe we can presume a closed system. Grandfather doesn't have a waste heat problem, he has exactly as much energy in his closed system as he WANTS, he just keeps 'recycling' it endlessly using perfectly efficient systems. Heat isn't 'created' energy, obviously, just transferred.
The same 'solution' can exist for a dyson sphere. The 'goal' is to close the system so that all the energy of the existing system can be captured and used for 'work'. Waste Heat may or MAY NOT be a problem depending upon the ability of the builders to use that energy in the efficient ways, and to capture and 'reuse' the waste heat energy. What is inside the Sphere, in fact what that energy is being used for, is just as important question as anything else.
I can postulate that anyone capable of building a Dyson Sphere, and more importantly with a use for that level of energy use, probably is no longer restricted by purely material sciences for their technology. By using powerful gravitic fields the captured energy never touches the shell of the sphere, waste heat from whatever it is they are doing is 'vented' right back into the center of the sphere, sort of like how the alternator on your car recharges the battery used to start said car... the shell of the sphere may be 'warm' in comparison to the rest of space (waste heat from the devices used to funnel the star's energy may be 'dumped' into the mass of the shell to radiate out, a minimal loss, but even that is unnecessary given the level of technology we are talking about.
If you are talking a purely closed system, then we might be speculating that the interior is something like a private reserve: the star is kept stable by the same forces that keep the shell from melting. The system is akin to a museum (or a shelter) for a very high TL race that wanted to remain hidden and thinks (thought?) in the very very long term. If the energy is being used, however, it is probably being 'shunted' somewhere, somehow. Maybe the entire sphere is nothing more than a stellar 'battery' used to 'power' a pocket universe, maybe the entire thing is a planet killing super laser (credomar!), taking all that captured and stored energy to make a ridiculously impractical interstellar beam weapon that can only fire once a millenium, but when it does it can crack a planet light years out (requires one hell of a targeting computer and a lot of patience......).
Okay... I'm done now... second time today my brain ran away with my fingers...

o: