2007 Mazda 3 - Compressor replaced, A/C runs then not

there is more to the a/c than just the compressor,when you engage the a/c one or more electric fans in the front of car kick in, have someone watch to see if they start up when you press a/c button. If this doesn't happen, check fuses for a/c fans, if they don't run you will get hot air. This is one scenario, irrespective of vehicle make.

does your blower work on all speeds if not it might be your a/c resistor

It will be the pressure switch is bad if it is cool for short period,then stop,s.it control,s the flow of freon,