I have been interested in making a turbine engine powered paramotor just to say that I did, but turbine engine use a ton of fuel and micro turbine engines like what we need have very short rebuild intervals.
I was thinking of using this jet cat turboshaft engine as it produces 15kw power (20hp) and it only weighs 10 pounds. A 9 liter fuel tank(about 2.5 gallons) would last only 90 minutes at an idle and 16 minutes at full throttle. So flight times would be really bad.
On the rotary engine topic, a few companies have offered wankel engine paramotors before. Parajet had a 40hp cyclone paramotor powered by a rotron engine, scout made a prototype 60 hp rotary paramotor, and on an old website flat top paramotors had listed for about $10k a rotary paramotor with 50hp. The problem was that they didn’t have that much more thrust than their piston powered counterparts, but they were all heavier, consumed a lot more fuel, and were more maintenance intensive.
I will be interested to see what happens with liquid piston as they claim to have worked out some of the issues, but I first started hearing about them like 5 years ago and last I heard they still didn’t have an engine in production.