How much would it cost to fix the back of my phone

Unsure really. At least in the past, any kind of case damage was assumed to cost Apple's "other damage" price, which is $549 plus tax in the United States regardless of the capacity. That would have been a complete replacement with a replacement device that could be new or refurbished. However, there are indications that iPhone models since the iPhone 12 might be repairable for just the glass. However, if there are any prices for that, Apple doesn't seem to publish them as they do for battery/front screen/other damage replacements.

lobsterghost1 wrote:

Actually, they do for iPhone 12 and 13 models.

Yes - I understand that the AppleCare+ repair deductible on those models for back glass damage is now the same as for the front screen assembly. However, I have yet to see a published out of pocket price for it without AppleCare+. I'd be interested.

Actually, they do for iPhone 12 and 13 models.

Start here.

iPhone Repair - Official Apple Support

Yes, I would as well.