Where do you get that idea? A typical EER 12 air conditioner will move 3.5x the heat energy that it consumes (COP = 3.5) [0].
At the location of the hvac it puts 130% of the source heat into the environment.
But 30% of that heat put into the environment came from electricity generated in a power plant. Power plants are typically less than 50% efficient, so it put's out as much heat into the environment at the source of the electricity. Bumping the value to 160% (130% + 30%).
However waste heat is a small fraction of the heating that electricity generation produces. Very roughly 10 times as much heat is trapped via the CO2 released than heat is released by the power plant. Bumping that value up to 460% (160% + 30% * 10).
I.e. 4.6 units of heat are put into the environment for every unit of heat removed from a closed system.
(Obviously the details of this depend dramatically on the environment. Heat pump efficiency depends on the degree of temperature gradient, CO2 release and power plant efficiency depends dramatically on where the power is coming from, which changes with where you are located.)
Also I think you're a bit pessimistic about modern power plant efficiency-- combined cycle plants do better than 50%, and that's before we're considering any benefit from renewables.
Fair point, I guess my argument makes more sense if we were discussing moving naturally occurring heat out (i.e. household ac) than with respect to cooling a datacenter.
Nitpicking the numbers used in the estimate... is probably not worth it. Every bit of it is a very rough order of magnitude number. If you're somewhere with 95% renewable energy it should be an order of magnitude better, if you're somewhere where energy production is dominated by an inefficient coal plant it should be an order of magnitude worse.
Wouldnt solar / Wind have a smaller CO2 foot print and hydro electric be more efficient ?
But without scientific sources, these are only wild speculations.
Consider, if you can achieve a fully passive cooling solution by dropping a datacenter into a lake, you've reduced the energy consumption in service of cooling by 100%.
(In reality, water cooling isn't "free," but I'm willing to bet the amount of energy required to dump heat into surrounding water is a whole lot less than the amount of energy spent for the compression cycles and forced air of above-ground HVAC systems. Water cooling using direct application of chilled water is already a thing, using lakes or retention ponds as places to dump heat; what being at the bottom of a lake gives you is a more consistent and proximate source of cool water than you might expect from a current chilled water distribution system)
Obviously this can be mitigated if you are able to get renewable power from a nearby source like a geothermal plant, hydroelectric, or solar. But if you are using fossil fuel power from a long distance away, that means any unit of heat moved by an HVAC involved many units of heat production to ultimately move that unit of heat.