Hello and welcome to the forum.
You could have a weak main fuse that when it heats up it opens and when it cools it makes contact.
You could also have a bad pulse generator that gets hot and the bike cuts out and refuses to start. I normally test them with 'Freeze Mist' for electronics, when the component is good and hot I spray the questionable part with this and if the bike starts, I found the problem.
That is correct, the GL1200 can run on two cylinders. My 86 Aspencade was running on only two cylinders when I bought it, the left cylinder head valves were bent causing no compression at all. The bike will run on two cylinders but not have any power to move it up the slightest incline, just level ground. At least that was my experience.
I know it's a year later, but maybe someone else has this problem. The same thing happened with my 1986 GL1200 Aspencade, would run for a while and then no ignition until it cooled down, tried coil pack, ignition module, drained fuel tank, replaced fuel pump. Turns out it was the crank sensor. A cheap part to replace, right on the front of the engine. Four wires connect up to where the fuze box connects on right side under tank covers. Hope this helps anyone with the same problem.