I noticed my A/C will be really cold when you turn it on and then after awhile it will start to blow warm air. When you turn it off and let it sit for a little bit, it will start working again but eventually stop.

Turn the AC on and open the hood and make sure both radiator fans are running.

They both run when A/C is on

What year and model?

Possible the charge is just low.

There are a few years that are having trouble with the compressor clutch not engaging properly. Basically the AC works for a little while then it stops.

It is a 2003 Subaru Legacy L, I checked the charge on the low side and it was in the lower 30s and it was 80 out.

*** When it starts blowing hot - I assume the compressor quits running?

 

That pressure looks roughly on target but I'm not a pressures expert.  The most common issue I see is they loose charge through leaky orings on the compressor:

http://www.subaruoutback.org/forums/99-do-yourself-illustrated-guides/43428-diy-c-air-conditioning-leak-refrigerant-repair-5-less-15-minutes-less.html

I don't see any compressor oil around any of the connections, that's the first thing I looked for when it started giving me trouble. The compressor does quit running when it blows warm air, I have to turn off the A/C for a few minutes and turn it back on and it will start running again.

The low pressure side only tells part of the story. And if you're using one of those little cans with the plastic guage those really aren't that accurate.

 

The 03 should run about 18-28 psi on the low side on a 90° day.

The expansion valve regulates the pressure drop from high to low and will usually maintain around 25-28 psi on the low side even if the charge is a little low. But if the charge is low the refrigerant will freeze the evaporator core and the system will turn off until the core gets back above a certain temperature.

 

If the system is turning off because the core is freezing, adding some refrigerant will fix that.

 

The other thing to check is to see of the compressor clutch is getting power when the system stops working. If the clutch is gettin power, but isn't engaged there's a problem with the clutch. It may need adjustment, which can be done with a special tool. Or you can replace the whole compressor.

 

If the clutch is NOT getting power, it's because of the safety switches that are on the system to shut if off either due to incorrect pressure or temperature.