The phrase "fearsome reputation" refers to having a strong and intimidating reputation. It means that someone or something is known for being feared or respected by others due to their power, abilities, or actions. Full definition
It has a very fearsome reputation as the best service dog for those in the military as well as in law enforcement. (mypetneedsthat.com)
The performances are all strong; especially good is Bullhead star Matthias Schoenaerts, rocking a flawless American accent to play a hoodlum whose fearsome reputation precedes him. (film.avclub.com)
Unfortunately, after starting with such promise, the months of September & October now appear to be living up to their more usual fearsome reputation. (wexboy.wordpress.com)