Azmodeus, aka Asmodeus, aka Asmodai, appears in a lot of mythologies, often as the ruler of Hell, and of course he does nothing to dispel this notion. He is also known as the demon in charge of lust, and this much is true (although Hell’s rulership is not so clear-cut that every Prince has a mortal sin, many do have a “sphere of influence”, ie Mephisto with temptation, Belial with disease, etc).
As such, Azmodeus is ambivalent to the power struggles of the other Princes. He is comfortable with so many mortals thinking him the “king of Hell”, and with his own work twisting sexual desires into uglier needs. Like Malefis and Astaroth, he is a loner, and keeps to his tower on a black sand beach on the edge of a corrupted ocean.
He is also Hell’s historian, and holds all kinds of knowledge on every incarnation of the hierarchy.