Stretching rubber or other hyperelastic material heats it; allowing it to relax cools it. If you have a set of concentric rubber tubes you can twist, with metal rings embedded at intervals in the tubes to keep them from collapsing, then the torsion should manifest as heat which could be carried away by a fluid passed through them; allowing the torsion to relax should cool the tubes, which can be used to cool a fluid passed through them at that time.
This allows the construction of a refrigerator whose refrigerant is solid and thus very safe. By passing the same air (or other fluid) through the tubes during successive cooling cycles, it should be possible to cool it to lower and lower temperatures, until whatever leakage is inherent in the system cancels it out.
The cooled, stretched rubber has lower tension than the stretched rubber did when warm, so the relaxation cycle returns less energy to the motor than was put into it originally, which is where the ability to reduce the entropy of the air comes from. Still, it’s likely to be significant, so it may be desirable to use a flywheel or something in harmonic motion to recover that energy for the next stretching cycle.
Fatigue is potentially a major problem for this device, since the rubber needs to last through at least tens of thousands of stretch-and-release cycles to be useful at all, if not millions; and the degree to which the rubber can be stretched safely determines the achievable ΔT.