Let's see how much energy you can retrieve from your room:
Let's imagine you have a 1000 ft² apt with 10 ft ceiling giving 10 000 ft³ of air (283 m³). The temperature is at 90°F (305 K) and you cool it down to 39°F (277 K), the temperature of the coldest heat sink I can't think off, i.e. deep underneath the ground or a large body of water.
The specific heat of air is 1.005 kJ/kg/K and its density is about 1.16 kg/m³. This gives you 1.005 * (305 - 277) * 1.16 * 283 = 9238 kJ of energy stored in your room that you can recover. But to recover that energy, you will need a heat engine. Maximum efficiency possible is the Carnot efficiency which is 1 - 277/305 = 9%. A more realistic maximum efficiency would be the endoreversible efficiency, which is 1 - sqrt(277/305) = 4.7% (see
heat engine efficiency for more info). Using this last efficiency, this means that you can expect recovering only 434 kJ of the 9238 kJ calculated earlier, or 120 W.h. This is the equivalent of filling a
12 V, 10 A.h, battery.
This is if you «empty» your apt once, say, at the end of the day. If you do so, the room should be at 39°F at the end of the process. But because the room cools down, the engine efficiency will drop as well, further reducing the energy you can recover.
If you do this during the day, while the sun, computer and stove heat your apt and you maintain the temperature at 90°F, how many batteries you can fill with your heat engine will solely depend on how much heat the sun and appliances produce in your apt. You could also keep your apt at a cooler constant 70°F as well, but in this case, you would have an even smaller efficiency for your heat engine.
Such an engine would probably be very costly, if feasible. Engines with small temperature differential cannot usually handle a lot of power, i.e. how much energy they can transform per unit of time. So it would probably take a very large engine to remove enough energy to cool your apt. Some engines that
might be used for this could be a
thermoelectric generator or a
Stirling engine.
A better way that is actually used with that concept is
geothermal cooling, which is a more efficient air conditioning (so it doesn't give you free work, but it costs less energy).