Ac starts blowing cold then in a short time quits blowing cold air

See this post on your exact problem http://ricksfreeautorepairadvice.com/car-ac-blows-cold-blows-warm/

TO MANY REASONS FOR IT TO QUITE WORKING IT WOULD BE BEST TO LET SOMEONE LOOK AT IT

fans are not running

check to see if ac clutch is turning off and on

Is your car overheating?

This is common problem,noticed on many other 200 chevy equinox car models too.Click this link below to follow troubleshooting procedure :----http://technoanswers.blogspot.in/2012/02/chevy-equinox-quits-heat-after-few.html
---------Thanks.

Check if the compressor is working.
looks like there is a problem with compressor , hence working intermittently.

Have you noticed that the AC quits blowing cold air when you are heavy on the throttle, or climbing a hill? Does it return to blowing cold air when you let back off the throttle and relieve the load from the engine?

If you answered YES to those questions, your AC is operating normally. AC compressors take quite a bit of power from the engine to operate. In vehicles with smaller engines, the AC compressor will disengage when there is a heavy demand for power placed on the engine, and you will not get cold air. When that demand is gone, it re-engages power to the AC compressor.

u have a temp door accuator motor shorting out

poss causes are, sticking low pressure switch, defective Ac compressor clutch closing coil or incorrect clutch driven plate air gap, spec is 0.26 thousands.