When replacing a horizontal output transformer, apply heat sink compound to the bottom of the transformer and it will help conduct heat from the transfromer to the chassis and the transformer will run cooler.

If so, probably not by much. The steel chassis has a lot of thermal mass, it's not just going to soak up heat from the small flyback frame like you would think. If it were made of aluminum, sure. For the most part, a correctly aligned horizontal oscillator and a small muffin fan will be more than enough to keep things happy. Plus it's easier than lifting the transformer off the chassis, and potentially damaging something in the process.

If it's running so hot that the core needs heat sinking, something's drastically amiss. :o