For example, suppose that in a troglodyte encounter, a PC fails his save versus the enemy's stench aura, and is nauseated for n rounds. On round n+1, does he need to reroll that save, or is he considered to be used to it by that time?
[RPG] Do aura-based effects require a new save every time they wear off
In Pathfinder, Stench has the universal rule including:
With nothing supporting that failed saves confers a similar immunity. Frightful presence has a similar rule to stench:
These are the only aura-type Universal monster rules that describe anything about immunity. Another aura-type ability, Fear Aura, says nothing about immunity. Thus, Fear Aura would only give immunity to itself if the monster's descriptive text explicitly grants it. Similarly, descriptive text in a monster entry may grant immunity to a creature's stench or frightful presence abilities.
Troglodytes have no such descriptive text, so a failed save causes the sickened condition for 10 rounds. Anytime after the 10th round where they are in the aura they must make a new save, where failure is another 10 rounds and success grants immunity for 24 hours.
As the question is tagged [Pathfinder] and [D&D 3.5], I checked the OGL stuff from 3.5 and found that none of the referenced wordings changed (Troglodyte's stench save text is word-for-word, as is the saving throw immunity for Frightful Presence)
In response to the titular question, Pathfinder's universal rules and the 3.5 equivalent rules state that, in general, yes, Aura-based effects require a new save every time they wear off.