Does it take more energy to maintain a higher cooking temperature?
The limiting case would be a pot that is only emitting energy via thermal radiation, because it is (in the imagination) perfectly insulated and so not losing heat into the air via escaping vapor through the lid, or conducting heat through the pot material into the air.

In that case, the energy loss from the pot is only that of the black body radiation, which is a function strictly of temperature.   Energy radiates away at a faster rate at higher temperatures.

Even if you relax these stringent imaginary conditions, all heat engines are based on a temperature difference.  All of the heat loss mechanisms on the pot (such as losing heat to vaporization, or conduction through the surface of the pot into the air) are also driven by temperature difference between the contents of the pot and the surrounding air.  The larger the difference, the faster the transfer on an absolute basis.

So yes, maintaining a higher temperature in equilibrium requires more energy per unit time than a lower one.   But you would need someone far better at physics than me to give you the absolute mathematical models.