It is said that ACs are counterproductive in fight against global warming, in that while they may make the local environment temporarily livable, the greenhouse gases produced while making the electricity needed to operate them heat up the rest of the Earth by much more than the relief from the AC itself. By how much exactly is that? Note that here I am interested in the global impact of greenhouse gases specifically, not in the local heat island effect (given how ACs do not destroy heat but only move it from inside to outside, and add extra heat from running the compressor itself). Let's also assume all electricity comes from fossil fuels (ACs might become a viable solution if 100% of AC electricity came from renewable solar, which is actually a reasonable goal to strive for given how both AC and solar are most active during the day, but at the moment most of electricity delivered to me specifically, for example, comes from natural gas.)

Here's my estimate. Let me know if it is reasonable! Methane has energy density of 891 kJ/mol, burnt into CO2 at 1 mol : 1 mol. Gas turbines have efficiency up to 60%. The radiative forcing of CO2 can be calculated as: ln(new ppm/old ppm)/ln(2)*3.7 W/m**2. For example the 131 ppm increase in CO2 since 1750 up to 411 ppm has a radiative forcing of 2.05 W/m**2 (is that across the entire Earth's surface? or only its crosssection?), and CO2 has persistence in atmosphere for at least 1000 years. The atmosphere composition is 78% nitrogen 21% oxygen 0.9% argon so its molar mass is:

.78 * 28 g/mol + .21*32 g/mol + .009*18 g/mol = 28.7 g/mol 

And total atmospheric mass:

4*3.14*(6.37e6 m)**2 * ~10000 kg/m**2 * 1000 g/kg / (28.7 g/mol) = 1.78e20 mol

Suppose 8 billion people each run 1kW AC for 1 year, with electricity from natural gas. (That's similar to our total current global energy consumption of 20TW, though of course we use power for things other than just AC or electricity, but also most energy comes from coal and gasoline not just gas, and 80% comes from fossil fuels not renewables.)

8e9 people * 1000 W/person * 60*60*24*365 s / (891e3 J/mol * 0.6) = 472e12 mol

That's 472 teramols of CO2 (20.8 gigatons) added to the atmosphere each year, or 472e12 / 1.78e20 * 1e6 = 2.65 ppm (parts per million). It is believable that having done so for a hundred years we have raised CO2 concentration from pre-industrial levels up to 411 ppm. The radiative forcing is:

ln((411 ppm + 2.65 ppm)/(411 ppm)) / ln(2) * 3.7 W/m**2 = 0.0343 W/m**2

Or for the whole earth:

4*3.14*(6.37e6 m)**2 * 0.0343 W/m**2 = 17.5 TW

What is my individual contribution for 1 hour?

17.5e12 W / 8e9 / (24*365) = 0.25 W

That is, if I run my 1kW air conditioner for 1 hour, the entire Earth will be solar heated by an extra 0.25 W for the next 1000 years. That doesn't sound like much, but it adds up over time: I spent one kilowatt-hour in one hour on cooling, but the rest of the Earth will be heated by an extra 0.25 W * 24*365 hours = 2.2 kilowatt-hours in the next year, and again every year thereafter. Multiply that by 8 billion people or a hundred years and it adds up a lot, even considering the heat is distributed across entire planet surface not just areas where people live.

So my answer is 1 kWh of cooling = 2.2 kWh of heating per year for the next 1000 years. By same calculation in terms of mass, 1 kg of CO2 = 7.4 kWh of heating for every year thereafter. Is this accurate?