newtonian mechanics - Does a rocket engine apply more power as the rocket's speed increases?

To clarify my level of knowledge, I'm a high school student all the way through AP Physics C: Mechanics.

So, let's imagine that there is a rocket travelling through the vacuum of space (ignoring gravity, air resistance, and all that). It's constantly accelerating with a $500\,{\rm kN}$ engine. Suppose, at some time, that it's going at $1\,{\rm km}\,{\rm s}^{-1}$. Since ${\rm power} = {\rm force} \cdot {\rm velocity}$, the power being applied instantaneously by the engine should be $500\,000\,{\rm kW}$, right?

Suppose that the rocket's speed has doubled to $2\,{\rm km}\,{\rm s}^{-1}$. Isn't the instantaneous power from the engine now $1\,000\,000\,{\rm kW}$?

So what confuses me is this: if the previous two paragraphs are correct, then isn't the engine using energy at a higher rate simply by virtue of going faster? And doesn't that mean that it's using up fuel at higher rate? But how could it be using more fuel when it's still applying the same force?