The most common and most likely answer to this crossword clue is the 7 letter word HAUTEUR
I've seen this crossword clue in The Guardian
H
A
U
T
E
U
R
More clues leading to the result HAUTEUR
- Hard film director showing pride
- Superior attitude of hot film director
- Arrogance of hotel director
- Gallic arrogance is hard on Balzac, perhaps
- Arrogance of Mann shunning human nature perversely
- Pride of person who detests being given uniform on two occasions
- Hot movie creator revealing arrogance
- Arrogance of Hollywood’s premier film-maker
- Hollywood's top film director's arrogance
Buddy explains!

The straight part
We know that "Gallic arrogance" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case gallic arrogance = hauteur.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does HAUTEUR mean?
HAUTEUR noun- Haughtiness or arrogance; loftiness.