The idea for using pressure to lower temperature is what your air conditioner or refridgerator uses. First you compress a gas. Using the PV=nRT formula, we use Kelvin because it's an absolute temperature, unlike Celsius or Fahrenheit. Or you could use the Rankin scale, which is also absolute, but most people are unfamiliar with that.
Anyway, room temperature is about 300 Kelvin. Put the squeeze on, say 1,000 atmospheres, and the temperature tries to rise to 300,000 Kelvin, 50 times hotter than the surface of the sun. Oh, wait a minute, you want to cool something off, not melt the emblem off of Superman's costume. Ok, let your squeezed gas sample stand still, radidiating all that heat until it returns to room temperature.
Uncork it.
The temperature of the gas snaps down a thousandth of room temperature, or 0.3 Kelvin, just three tenths of a degree above absolute zero.
You won't actually get all the way to absolute zero with this method, but you can get close.