Can a pulse jet be used on a light GA aircraft?
In principle yes. But some details will turn most potential operators off.

Noise is the obvious first one. You might not mind, but cockpit noise and vibration in operation will certainly put an unacceptable stress on the pilot. Just think how you will communicate with ATC or with a co-pilot. Also, your choice of airports to operate from will be rather restricted.

Vibration also means that you need to ruggedise the airframe. This will certainly eat up some of the mass benefits of the pulse jet, but is hard to quantify.

Now fuel consumption: The AS-014 would consume 350 g of fuel per hour for every Newton of thrust. Compare this to the 60 g/Nh of a modern turbofan. The additional fuel will eat up the remaining mass advantage once you desire to fly for more than just a few minutes.

Next will be take-off distance. Unless you use a catapult, you will need a long runway. The pulse jet has limited static thrust, so your initial acceleration will be poor. At the high end of the speed envelope, though, thrust should be plentiful.

Cooling, however, is no problem. Just expose the pulse jet to the elements and you are all set.

To summarise: Yes, a pulsejet is a possible power source for a GA airplane, but is a poor choice for a multitude of reasons. It makes more sense in an unmanned, one-way application.