What? Can we start with an example such as:
1) Take a room starting at 20C, how much energy to get the room to 25C?
2) Conversely take a room at 25C, how much energy to get the room to 20C?
Intuitively heating should be cheaper, since loses are heat. Whereas with cooling you'll have inefficiency (e.g. in the pump) in the form of heat, therefore more energy to remove that additional heat.
A typical AC can take 1 kW of input power and remove ~3kW of heat energy from the room. The outdoor part of this system is heated by 4 kW, because you put the high-power stuff outside, where it's unimportant to performance. You can't thermondynamically make an AC work if it's purely in the room.
With resistive electric heating, you put in 1 kW of heat and the room temperature increases by 1 kW of heat energy.
With a heat pump in heating mode, you can put the inefficiency to use, using a 1 kW input to extract 3 kW of energy from, say, the ground, and putting 4 kW of energy into the room.
Two main problems with the argument: Cooling, as you say, is typically 30->25. Heating is often 0-25, requiring 5 times more energy (though we should be doing this in Kelvin...) Second, heating is super easy to do thermodynamically by just burning dirt-cheap natural gas. For a Michigan anecdote, I paid about $150/mo to keep my house cool in July, and it's typically $100 to keep my house warm in January.
The air outside right now where I am is 0.7 degrees, I have my heat pump set to 21 degrees.
What the parent comment neglected to mention is the working temperature of the gas / fluid in the heat pump.
R410a boils at -48.5 degrees[1]. So based on the size of the system and some maths I couldn’t explain so I’ll hand wave away, there’s your answer. Sorry, that’s the best I can do, you’d have to read some Wikipedia entries to get a better understanding.
Modern heat pumps / AC can be up to 1:4.5 thermal efficiency, but it depends on the the temperature difference, the size of the system, and what you’ve got it set to.
Other places I’ve lived it gets to 48 degrees outside and you set the AC to 17 degreee because Australians are really bad at insulating their homes, so a 31 degree temperature difference, so in that sense the parent comment is flawed.
> You can't thermondynamically make an AC work if it's purely in the room.
We have two AC units in our home that are entirely inside, there is just a hose connecting the unit to the window and the hot air is blown through the hose.
By "purely in the room" I imagine they meant within a closed system. By having a hose connecting your units to the outside world your room is no longer a closed system (ignoring things like drafts and imperfect insulation etc that already prevented it from being considered closed).
The way modern heat pumps work, I can't answer your question yet. What's the outside temperature? Let's say that taking your room from 15C to 20C takes 1.0 energy unit. If it's 15C outside, then I can use a heat pump to gain a bunch of efficiency, say only using 0.2 units of electricity to heat up the inside by one unit. If it's 25C outside, I can do the same in reverse, but I do have to also cool the A/C. We'll say taking the room from 25C to 20C when it's 25C outside takes 0.25 units.
The problem happens when there is a big difference in outside temperature. All of the gains of heat pumps disappear. So cooling from 25C inside to 20C inside when it's 30C outside might take 0.5 units of energy, which is a pretty common case. Heaters generally have to deal with much bigger swings, though. It's not uncommon to heat from 15C inside to 20C inside while the outside is at -10C, which takes (asymptotically approaching) 1.0 energy unit.
So the inefficiency of the A/C producing heat does matter, but not as much as the difference between the desired temperature and the outside temperature. The rate of heat transfer of the house also depends on the difference in temperature. Keeping a house 10C cooler than ambient takes less energy than keeping a house 20C hotter than ambient.
Considering that temperatures on Earth vary from about -90C to 50C, and humans like to keep indoor temps around 20C, cooling is generally more efficient than heating.
Intuitively heating should be cheaper, since loses are heat. Whereas with cooling you'll have inefficiency (e.g. in the pump) in the form of heat, therefore more energy to remove that additional heat.