Since you mentioned it does it hot or cold: worth also noting whether it does it after the car has been sitting for days versus after a short stop.
A fault that only shows up on the first start of the day points at something that settles or dries out. A fault that is equally likely on the tenth start points at pure mechanical wear. You have already established temperature does not matter, which rules out a few things: adding the sitting-time observation costs you nothing and rules out a few more.
Write it down for a week. Diagnosing from memory is how people end up replacing three parts.