Welcome to 2CarPros. This isn't a typical problem and could be caused by a few things. It could be a bad solder joint in the security module that moves when it gets hot or it could be a bad transponder antenna coil acting the same way. Unfortunately about the only way to find out which would be to use a VAG type scan tool that could watch which item it failing when it gets hot.
About the only DIY test I can think of would be to get some "canned air" it is actually a refrigerant, that you could use to spray around the key area when you have this problem to see if cooling the transponder antenna makes the light go out faster. If it did then you would have an idea of where the problem is. Otherwise without a scan tool it would be a real goose chase.
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Thursday, October 31st, 2019 AT 3:58 PM