Can the temperature gradient between an aircon's hot output and air outside do work and save part of aircon energy? I usually see air conditioners releasing hot air from the inside to the world outside. But since it is warmer than the air outside, I cannot see why this gradient could not do some work and possibly save some energy, like reducing the aircon's electricity consumption.
 A: You need to look at how hot air is blown from the external radiator-it is due to a fan. The natural heat currents generated from temp difference  are not enough to cool the radiator, this is why fans are installed in it .
If you try utilizing the air flow significantly it would restrict air flow around the radiator and  decrease the efficiency of your air conditioner.
Also scale and cost matter too.
There is a data centre that warms a swimming pool from the heat it generates but are you going to try doing this with your CPU?
A: Suppose it's wind still outside. Of course, heat always flows from hot to colder regions. So, in theory, it's possible to let the heat do work and you can get some energy that you invested in the aircon back. In what way is of no importance.
It might be clear that you can't use the energy completely so the aircon will keep conditioning air forever. In that case, you would have created a Perpetuum Mobile, which according to the second law of thermodynamics can't exist. There will always be energy lost in the form of heat.
A: The compressor has just consumed electrical energy in order to heat up the refrigerant.  Instead of grabbing the energy from the gradient, wouldn't it be better to just not heat it up so much?
The temperature that the condenser reaches over ambient (or the split) is what is required to reject the load from inside.  Given a particular condenser and a load, the split will rise until the difference is sufficient to reject all of the heat.
If you now attempt to extract work from that difference, it's like putting an insulator around the condenser.  Yes, you can get a little bit of energy back, but in response the compressor will do more work (and consume more energy).
