Frank was dismissed by Spurs after a woeful run of results left the North London club hovering five points above the relegation zone in 16th, with the Danish coach unable to replicate his success at Brentford.
Arteta said Frank's sacking was "very sad news" but acknowledged the brutal nature of Premier League management.
"Thomas is an excellent coach. He is an extraordinary man as well and he has proven that in the Premier League," Arteta told reporters ahead of Thursday's Premier League trip to Brentford.
"We know our responsibility is beyond just performance. Sometimes results dictate what happens to us. I wish him all the very best for whatever he wants to do next."
Spurs next face Arsenal on February 22 in the North London derby with either an interim manager or a permanent appointment in charge, adding another layer of unpredictability to the fixture.
Arteta revealed his team has become accustomed to facing opponents under new management this season, forcing Arsenal to adapt their tactical preparations and focus on themselves rather than second-guess their opponents' approach.
"At the end you learn the lesson not only with the change of manager but with what you expect from a manager to do against you as well, because they change it a lot this season," Arteta said.
"So, focus more on us and make sure that what we want to achieve... The set-up is one thing, the player profile is another thing and the momentum of the game and context is something else.
When asked if Premier League clubs were being too impatient, Arteta said: "In the context of every club it is very different, but it's always a possibility.
"I can just talk about Frank and what I know as a coach, because I faced him many times and how he prepares his teams and what he did at Brentford too.
"But in the end this league is so competitive. We are all vulnerable because anybody can beat you on the day and that's really tough to manage."
Arteta said the club would take late calls on the fitness of Bukayo Saka and Martin Odegaard while Max Dowman is still unavailable.