Yea, there is no "thermodynamic reason" for a limited supply of energy. Even if we started expending too much energy in the atmosphere (say if we found an efficient way to fusion), we could pump this excess heat to space, if it would ever come to such an extreme.
I see that none of you have heard of Carnot efficiency, or even the second law of thermodynamics. Even if we were to get electric energy from magical pink unicorns that produced no heat, the usage of that energy would produce waste heat. And no, you can't just pump heat into space; if we could that would easily solve global warming.
Constructive proof: build a large suspended reflector (mirror), put you power source on it and pump heat from earth to a sphere of any material, which radiates heat into space.
No, this of course can't easily solve global warming due to enormous cost it would have and ludicrous efficiency. It's way cheaper to just absorb less heat from the sun (less CO2).
Sometimes it's best to do the math before claiming stuff. Now, using thermal radiation from a metal puts your emission spectrum at blackbody radiation around or below 2000 Kelvin, and in that band around 50% of your emitted energy will be absorbed by the atmosphere (thus will not leave earth). Your heat pump going from 293 K to 2000 K will have a laughably low Carnot efficiency (the highest efficiency allowed by thermodynamics): for every Joule of heat you pump from the earth to the sphere, you have to use more than 20 Joules from our magical unicorns. Of these 20 Joules, 10 will be absorbed by the atmosphere and heat the earth. But you have only removed one Joule of heat from the earth. Thus your apparatus is not able to cool the earth, in fact it will be heating the earth by a lot.
Alright, thermodynamics + not requiring magic (except the unicorns) then. Thermodynamics has everything to do with it; if it were not for the Carnot limit on heat pump efficiency, your proposed scheme for heat removal would work fine. If it were not for the Carnot limit, we would have no problem growing global energy consumption at 2-3% per year (i.e. exponentially) for a hundred more years.
So, if you put your apparatus anywhere but in the low atmosphere, how do you expect it to pump heat from the low atmosphere (where the heat is)?
I'm not on any high horse, but I'm surprised at how quickly people dismiss this problem as absurd. It's quite obvious if you know some engineering thermodynamics. I think people dismiss it out of cognitive dissonance, since they don't want another unfixable world problem.
1) Thermodynamics imposes a finite limit on the energy consumption on Earth;
Sure, it might be necessary to build a fleet of titanic cooling towers to cool the atmosphere, I don't think we will ever need to do that; but thermodynamics doesn't forbid cooling Earth, hence your claim is false.
That out of the way
2) The direct thermal output of power sources/uses is going to be a problem in the near future
I claim that's also false. In fact, the burden of the proof should be yours when you make such a claim, but let me show why I believe otherwise: (approximate figure from wiki [1])
142 PWh of energy was used worldwide. Solar irradiance is 340000 PWh. It would take almost 200 years of steady 3% growth (or a 240x increase in consumption) to even reach 1% of solar irradiance as internal heat. Even then, the impact of greenhouse gases massively overshadows that.
On top of that, there's no activity requiring exponentially more energy, or evidence that we may continue expanding energy demand for hundreds or years. In fact, if you look at per capita energy consumption vs GDP [2], you can clearly see a saturation of energy needs. Once the developing nations reach this, it seems demand growth is going to slow down. Nothing physical ever maintains an exponential growth for too long.
