FIA designs a mandatary "Driner Protection Capsule" and telemertry system capable of measuring real time horsepower. Only two rules; cars must use both and produce no more than 800 HP period. Everything else is legal.

yea.... you wish!

Can I use the 800hp engine to power the pumps and systems needed to administrate the liquid fuel rocket?

as long as the rocket does not have more than 800hp.. yes.

Rockets have thrust.

Power is power no matter what.

A power restriction would be meaningless. You need a torque restriction

What - no gearboxes?

The equivalent power of a jet or rocket engine depends on the speed of the vehicle. At 0km/h and full thrust a jet or rocket engine effectively has no power.

Neither does a piston-engined car.

True, but the engine has a maximum power rating. Jets and rockets aren't rated for power, but thrust.

Thrust is force, horsepower is work. Make sure your rulebook clarifies all terms it uses.

Power is rate of work.

See, it's important to be specific

Thats where many fall, you can't cover everything as clever F1 Engineers keep proving, simple singular rules without interpretation and strong management solves the problem.

Only because of an arbitrary decision taken at some point. Thrust is an important criterion for rating reaction engines, due to the lack of a gearbox or variable pitch airscrew to multiply force or torque. However the product of thrust and exhaust velocity (power) is also an essential piece of information as thrust alone does not define the engine's suitability for a particular application (subsonic, supersonic etc)