Bottom line: 40% efficiency, which is better than I expected but the competition is batteries at 80+% efficiency. It's a hard sell, especially as continual improvements in battery storage will continue to eat away at their niche.

5,000 W/kg sounds great on paper compared to 150 W/kg for batteries and is even in the same ballpark as gasoline at 12,000 W/kg, but I think that's just the figure for the fuel. I don't think it includes storage, the solar panels, the burner, etc... The cost is an open ended question as well. Maybe this will pan out for aircraft?

The gasoline vs H2 ballpark is a little wider because storage is not trivial for H2 -- you need to carry around a cryogenic and/or high pressure vessel instead of a plastic box -- which will detract from your p/w ratio. It also wants to leak out, so H2 is maybe better for fleet vehicle applications where they can refill daily. Granted, anything is better than burning more hydrocarbons!

If that is 40% efficient as in 40% of the theoretical energy input comes out as electricity then it's quite incredible but I find that hard to believe. It would put it in the same range as diesel engines.

The better comparison is Fuel Cells and vehicle based electrical generators. So you could put this in a vehicle or remote location, run it off hydrogen or natural gas, and get better efficiency. Potentially this could be a much better option for longer term storage in remote areas as well, where excess solar/wind could be used to crack hydrogen which then gets stored and later burned in one of these instead of a much much larger battery installation.

Rechargeables/battery packs have inefficiencies due to the grid and/or solar cells though, in terms of where to measure inefficiency?

It might not be a hard sell compared to home generators. Forget hydrogen. Think natgas.

Do you mean watts or watt-hours?