At pressures up to a few atmospheres most gases depart little from ideal gas behaviour. This answer is based on the ideal gas assumption.
I'll Suppose that you double the pressure by pumping more gas into the same container. If you then seal the container and leave it, the pressure is likely to drop, because most methods of pumping raise the temperature of the gas, and this will gradually fall as heat is lost to the surroundings. If you pump in more gas until the pressure is double at the same temperature as the original gas in the container then the energy stored will indeed be double.
This because the pressure p of a volume V of gas at kelvin temperature T is given by$$pV=nRT$$
in which n is the number of moles of gas and $R$ is a constant. So by doubling the pressure, keeping temperature and volume the same, you have doubled the number of moles. But the internal energy of the gas is given by $$U=n c_v T$$
in which $c_v$ is a roughly constant for a gas (except at very low temperatures). So doubling the number of moles doubles the stored energy if the temperature is the same. This makes sense because at a given temperature each molecule has the same mean (kinetic) energy