Many years ego, Earth was hot. Over time, it has lost energy and has become colder. Is it now in equilibrium or is its total energy changing?
"Total energy of the Earth" is somewhat of an odd concept, but there's no reason we can't really entertain it. It brings up some genuinely difficult questions. The right way to approach this is to define the system correctly and then identify forms of energy content and flows.
Things to "count" in the Earth's energy:
As I look at this list, I believe that all of them are steadily decreasing. Nuclear decays in the Earth's core continue over time, and this converts nuclear energy into heat content. I've heard that nuclear decay comprises a large fraction of the geothermal energy conducting through the crust, so it follows that the nuclear energy is declining at a similar rate as the heat content.
The rotational energy is constantly being transferred to the moon slowly, and this is similar to the maximum theoretical tidal energy that could be extracted.
Heat energy, of course, is lost by blackbody radiation to space. Global warming "blankets" our planet a little more, so it would initially decrease this. However, the heat content of the oceans and biosphere (which have the capability to absorb this energy) are small compared to the total Earth. There has always been a deficit between Earth's radiated energy and the sun's incoming energy which is from the nuclear and thermal energy of the planet.
Earth has always been on-net losing energy to space by radiation. An increased greenhouse gas could theoretically change this, and cause the Earth to keep more of its heat. However, it's small compared to Earth's natural flows. Perhaps after some past super-volcano the Earth temporarily gained energy. However, that's certainly not the case today.
I will point out a slight fallacy in the question:
If normal heat content was the only store of energy, this would be a logical connection. Losing energy would mean lowering temperature. However, other stores of energy are present, notably rotational, nuclear, and gravitational. The conversion of these into thermal energy is strange. Rotational turns into tidal heating, and then contributes to the radiation deficit.
Nonetheless, if the Earth is taken as a whole, only the large center should matter significantly. That has almost certainly gotten cooler over time, in addition to releasing stored nuclear energy. If nuclear heat production was large enough, or if the mantle was insulating enough, it could have increased temperature because the heat from nuclear decay wasn't dispelled to the surface fast enough.
Venus, for instance, may have had cycles where the planet stored up extra energy, and then an event where the mantle went through a massive volcanic shift.
The heat generated from the Earth's core is about 4x10^13 W while the Sun provides about 1.7x10^17W so although the Earth's core is slowly cooling this has very little effect on the Earth's temperature.
The Earth is in equilibrium between the energy received from the sun and the energy it emits into space. If the amount received changes, then temperature of the Earth will change to keep equilibrium - whether this is due to the sun emitting more or a more greenhouse atmosphere absorbing more.
I'd say depends on the definition of 'total energy' - see Alan's answer above.
The total thermal energy is not in equilibrium, it is increasing: global warming, that is an imbalance of around 0.5 W/m^2, corresponding to a total imbalance of 2.5x10^14 watts (if I did the multiplication correctly...)
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