2012 Toyota Corolla with 1.8 liter engine, automatic transmission, 100000 miles runs rough when cold but not after warm up.
A lot of things can cause that. What troubleshooting procedures have been done so far? Have you performed a diagnostic scan for codes and live data? That would be a good place to start.
Also please note for future reference your question belongs in the body of your post, not the title line which is meant for a short description. Thank you. (Title edited.)

runs rough when cold but not after warm up.
Well that sounds like a fuel mixture problem of some sort. Most likely not enough fuel. Could be a lot of things. Worn/dirt injectors. Failing fuel pump. intake leaks. exhaust leak... etcetera. Diagnostics are the best way forward. Measuring fuel pressure with a gauge. Checking fuel trims with a diagnostic tool. Just looking for clues to point in the right direction.