Yes Benny, there are 2 carbon brushes on the motor. One you can see in pic #2. The red wire is attached to the copper lead. There will be one the opposite side just like it, probably with a different colour wire attached to it. If your router has seen a lot of use, they could be worn down and cause excess heat as Howard (Dutchman 46) suggested. Not difficult to check, or change.
Sometimes I wonder whether the world is being run by smart people who are putting us on
or by imbeciles who really mean it.