The micro APU idea has been a dream for a very long time. Nevil Shute (Norway) wrote about it in his lovely model engineering novel "Trustee from the toolroom". Isaac Asimov has a micro APU of prodigious power in "Foundation and Empire" that welds itself into a solid block of metal in the event of tampering. And Scott Crossfield writes eloquently about the tribulations of the micro turbine APU's in the X-15 that would seize solid inches away from his head in "Always another dawn". But despite this, for 50 years or more, the micro APU has remained elusive.
One reason is certainly materials science - bearing cooling being one and demagnetisation being another, both related to the immense energy density required in these hypothetical machines. Even the tiniest inefficiency, converted into heat as it must be, will push the machine past it's materials limits. Perhaps these restrictions will be overcome, but if they are, I suspect there will be many more pressing (and financially rewarding) applications than private aeroplanes.
I'd love to see a successful micro APU giving perhaps 100KW for maybe 20Kg or something in place of diesels weighing 1/2 ton or more that never seem to work when needed. There would be plenty of server centres, hospitals, and so in line for that.