Can't move shift lever in very cold weather.

What is your question?

When the overnight temperature drops below about 32 degrees, we can't depress the button on the shift stalk to allow us to move from Park to Reverse. When we step on the brake pedal, the rear lights do go on, which I think indicates that the switch on the brake pedal itself is working properly (?) and should allow the button to be depressed. The only remedy thus far has been to start the engine and allow the car to warm up for about 10 minutes. Then the button can be depressed. BUT this is NOT very environmentally friendly. What is needed and how much is it likely to cost?