A full bottle of CO2 is mostly liquid.
A room temperature it will have vapor pressure of about 800 psi.
In the fridge, the vapor pressure is about 600 psi.
The regulator has atmospheric pressure, bottle pressure, and keg pressure.
It will allow CO2 to flow based on the difference between atmospheric pressure and keg pressure.
If you move the regulator from sea level to a mountain top, the pressure supplied to the keg will change.
But it isn't affected by temperature.