I have an 10-year-old American Standard Freedom 90 HVAC system, with an outdoor AC compressor. My thermostat (Honeywell RTC6580WF 1001) stopped being able to turn the AC on (it could still turn the fan and heat on), so I called out an HVAC guy. He couldn't find anything wrong with the contactor (but he replaced it with a sealed one) or the cap on the compressor, and upon further investigation he was able to get the AC to start by manually shorting the R, G, and Y wires behind the thermostat. From that he determined that the thermostat was bad.
I replaced the thermostat with a new one (same model), but when it tries to start the AC it immediately turns off for a few seconds (it's powered by the C wire) and then turns back on. It sits there blinking that it's waiting for the compressor protection timer, and then after 5 minutes it tries to start the AC again and reboots again and the cycle repeats.
I pulled it back off the wall and tested the voltages, and I'm getting a little over 26V between R and all the other wires. However, with the thermostat off, if I short R and Y, it now blows the 5A fuse on the furnace's control board. I've tried this twice, with the same result.
EDIT: I think I found the problem: https://photos.app.goo.gl/yeGaGdQPyAbwvw2x8
I heard scurrying when I first approached the compressor, so it's likely the cable became rodent food at some point. It's too far back to splice inside the compressor housing, so I am going to either need to figure out how to do a weatherproof splice or splice it inside the house and run a new wire through the wall.