L'événement a lieu quatre fois ____.
Should "l'an" or "par an" go into the blank, or is it exactly the same either way? Is one more formal than the other?
Both are correct, but "deux fois l'an" is pretty aged. You can hear it from grandparents or people living in the countryside.
Prefer "Deux fois par an", just know that the other one exists.
L'événement a lieu quatre fois l'an.
is equal to
L'événement a lieu quatre fois par an.
in periodicity.
The first has a light negative overtone as something not very reliable or not very important while the second is just fact.