Granted, if we did have the equivalent energy of a sun - we are unlikely to be spending it all on earth. Escaping earths gravity would be a trivial expense.
You can't just put the power plant in space and keep the heat from generating the power away from the earth, because the heat gets created where the power is used to perform work. You have to avoid consuming any of that power on the earth entirely.
And you can't just beam the waste heat from every process away from the surface of the earth, that violates the second law of thermodynamics. The heat exists because the energy performed work, you are keeping the energy organized by beaming it in any direction, so it is not waste heat, it is direct infrared from the energy you've produced. You can only do this by not actually using the energy for work. This makes the whole exercise of even producing the energy pointless.
I think the biggest hole in the argument is the jump to 100% efficiency. we don't consume mor energy because we feel like it, we consume it because we need it, and a factor of 5 increase in efficiency would equate to a factor of 5 decrease in production of energy resources.
Surely this can't be right, because if it were, air conditioning (the normal everyday kind) would be impossible, right?
What we're talking about here is loosely analogous to air conditioning the entire Earth (with the heat being sent to space rather than the outside air). Sounds like a hard engineering problem, but not impossible.
Generating economic value from work that the OP describes would not be "air conditioning" it would be converting heat into work -- which cannot be over 100% efficient, and in fact would be significantly less than 100% efficient at the temperatures we know.
Removing that heat using a heat pump would require some off-world reservoir of cold, which doesn't exist. The reservoir of cold that we are using now (and in any conceivable future) is the earth itself, which is what the parent comment means by "you can't use the work."
Space has an average temperature of 2.7K, and there's a lot of it. Can interplanetary space be an off-world reservoir of cold?
It's possible to make radiative coolers that do exactly this, within limits. It's easy at night; harder in the daytime. Here's an example.
(This is also why cloudy nights are warmer than clear nights in winter--clouds prevent some of the ground's heat from radiating away.)
The technique has been used for centuries to make ice in the desert, but it only works at night and it's inefficient.
https://www.fieldstudyoftheworld.com/persian-ice-house-how-m...
With modern technology it's more efficient and it appears it can even be made to work in the daytime:
https://energyindemand.com/2020/10/17/generations-after-peop...
An air conditioner performs the work of moving heat from one place into another. There's a reason your compressor is outside, that's because if it were inside the heat pump wouldn't work. You're expending energy to move heat generating more heat, you'd just heat up your house. The compressor wouldn't work outside if the heat wasn't convected or conducted off the radiator into the air, there has to be a temperature differential between the ambient environment and the radiator.
With the earth, there's nowhere to pump the heat to, there's only vacuum. You'd have to convert the heat to something else like light, radiate it out, expend energy to do it, and in the process generate more waste what from that energy that will then radiate out at the natural pace. At best in this process, you'd break even and just be really expending energy beaming light out into space and accomplish nothing with regard to the heat you're trying to move.
Keep in mind that air conditioners are net generators of heat. The work they do moving heat around generates more heat. If you build a sealed sphere 100 feet in diameter around an air conditioner and plug it in, the air inside the sphere will heat up. This is why you cannot just open your refrigerator door to cool your house. Earth's atmosphere is warmer today around cities where a lot of air conditioners are in use. (Although excess CO2 has a much bigger effect of course.)
I am afraid it is you who misunderstand thermodynamics, my friend. :-) Entropy is a matter of degree, and as long as the beam leaving the Earth is less ordered than the beam coming down, it can work. For example, imagine a hot nail on the moon suspended a few inches above a mirror. The heat will radiate into space and the moon won't get hotter.
* Total energy consumption = 10^26 watts (hope the caret comes through HN formatting)
* Assume some magical generation and transport mechanisms which don't involve energy gradiants (avoiding pesky thermodynamic realities), and are 99.9% efficient.
* Total waste energy is 10^23 watts.
* Radiators are on Earth, and we dedicate 1/2 of the entire surface area (not just land) to radiators. Earth has about 5.1 x 10^8 km^2, so say 2.5 x 10^14 sq. meters.
* Radiators for the waste heat would be dumping 10^23 W / 2.5 x 10^14 m^2 = 4 x 10^8 watts per square meter. That's 400 megawatts. Per square meter.
* That's a black-body temperature of 9,165 K (the Sun's surface is 5,772 K). Peak wavelength is 316 nm. So half the Earth's surface is a glowing UV light pointed at the sky. I don't think we'd have to worry about the ozone layer disappearing. (A very bright, inefficient UV lamp--plenty of infrared, hard UV, and soft X-rays to go around!)
* OK, so put the power plants in geostationary orbit. Say 1 million power stations, each with radiators 1 square kilometer.
* 1e6 stations * 1e9 square meters = 1e15 sq meters.
* So, that's only 1e8 watts/meter! Hmm. Still 6,480 K, but peak is blue-violet rather than UV. Still lots of UV and infrared, but it's space, who cares. Just don't cross the beams (ha ha, meaning no deep-space traffic). And mind the Moon! (Wouldn't want to give anbody on earth a reflected sunburn).
* The radiation pressure from the waste engery for one power stations would be about 730 kN (160,000 pounds). Would need counter-balanced beams.
* Geostationary orbit is about 26,000 miles. The surface area of a sphere that big is 2e16 square meters. So about 1/20th of the sky would be covered with powerplant radiators.
And this is just for the hypothetical 0.1% waste energy. Multiply all of the above by 1,000 for the actual used energy. So you'd have half the Earth's surface as power receivers slurping up 400 gigawatts/square meter of X-rays and hard UV. Each power station would see a force of 739 mega-newtons; you'd have to have a counter-balancing beam pointing away from Earth. And then you'd need to radiate away all that energy after it was used.
Edit: Oh, and while 1/2 of the Earth is receiving 400 GW/m^2, the other half is using that amount. Which would be the real limit. I don't think anybody would carry around a 10 GW iPhone 237+.
Assuming we’ve done all of that… you would still need to deal with the rocket engine you’ve created. Sending out a tight beam of a suns worth of energy is equivalent to a rocket engine with 1x10^18 newtons, creating challenges of its own - including an appreciable acceleration of the planet earth.
A more plausible escape hatch for the energy may be a black hole heat sink. In theory, such a device could perfectly “recycle” waste heat into usable energy.