Mine did that when it was 15°. My mother in law lives in the basement and nearly froze. Had to relocate her upstairs to my dismay.The good news is your heat pump decided to give you problems when it is 65º not 15º.
My disconnect fuse blew and melted a part of the fuse holder. The problem was the disconnect fuse box I had is no longer common, so ordering a direct replacement was a bitch.Ya if you’re smelling something burning, Id bet with acman that your motor bushings are shot or the capacitor blew and the motor is overheating from not spinning. Relatively cheap fix either way.
My basement dipped to 52 before we got it back on. I paid about the same...We had a similar issue (without the burning smell) when it was 6 outside this week. Luckily mine worked good enough to keep it 63 in the House. Ended up being the disconnect. $88 and we are back in business. Hope yours is cheap!
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Good luck OP...