Does how hot an amp runs have any effect on sound quality ?A definitive answer can’t be provided without knowing if how hot the amp is running is normal or abnormal for the specific design. If it is running abnormally hot for some reason, in other words hotter than it was designed to run, then yes, its sound quality may be affected. By both whatever is causing the abnormal temperature and by the abnormal temperature itself.
The others have made very commendable attempts to answer your question as thoroughly as possible, especially in the post just above, but we might be able to provide answers that are more focused if you would tell us what the model number of the amp is.
Regards,
-- Al