The energy output is due to the gas giant contracting due to gravity. But there is no fusion taking place. Look up “brown dwarf”. It’s the same reason why brown dwarves (with insufficient mass to start the proton-proton reaction) have “surface” temperatures” in the neighborhood of 1200 K. Eventually the contraction will end due to equilibrium and the body will cool due to thermal radiation.
Keep dancing. That explanation doesn't hold water. Even on the sun it doesn't work. Solar "surface" chromosphere temperature averages 5800K—it is even cooler as seen below when measure through sunspots—but the upper atmosphere of the Sun is measured at from 1,000,000 K to 2,000,000 K. This is not the result of gravity with convection somehow bringing the core heat outward. One can not get a higher temperature in a larger volume with convection, it can only get cooler. It is certainly not going to get hotter by three orders of magnitude. Something other than gravity and convection is at work driving this phenomenon.