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Looking at building an old character in PF2e that was a cold sorcerer, but their powers came from a divine connection. The coolest, pun intended, class i can find that is similar is the Oracle, specifically the Flames Mystery. How much of the math do you think it breaks if we just cross out fire and write in cold for all the effects? Instead of seeing flames and smoke everywhere, she as the daughter of winter would see frost and snow obscuring her vision. The heat shield becomes a crippling chill around her. Fire and Sun domain become Air and Cold. Thematically this seems perfect, but i am not sure if the energy changes makes it more powerful. Fire is resisted alot, but cold is as well. Any thoughts?
I'd say that fire to cold is pretty neutral all things considered - shouldn't cause too many issues.
If you decide you want to add any slowing stuff to the now-cold stuff, I'd consider dropping a die size, but otherwise I think you're fine.
I think fire resistance is probably slightly more common than cold resistance? But I think I also see more fire weaknesses, and I certainly don't think my flame Oracle has run into a lot of problems with resistance anyway.
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Fire and cold seem like a relatively even trade.
Swapping either for acid or electricity seems like it might be more questionable, as resistances and vulnerabilities to those two are rarer (and sonic or force, definitely!), but I think cold for fire should be fine, assuming that you aren't headed into a specifically cold-vulnerable adventure like a modern take on Against the Fire Giants or something.
Outside a few of the more esoteric damage types, it doesn't really appear that Paizo really uses damage type as a balancing feature, so generally I think any permanent swap as part of creating a theme is probably going to be okay.