Fair enough.
Despite the fact that Harun is comfortable everywhere, he is very much an Al-Badia. He views everyone as being equal, and silently snickers at those who believe themselves to be better than others. He extends this to Sheiks and perhaps even his own cousin.
So he finds their dirt and exploits it. Not because he's evil, or because he wants to be a vigilante, he just thinks its funny for men to find out they aren't all they think they are.
So he snoops. He'd go by the Harem, finding out how they are treated. He'd check out the Sheik's room, to check to see if he finds any incriminating documents or signs of treachery against the Grand Caliph, or even just to see what plots the man might have, even if they're not evil plots. Harun likes to know what's going on.
It's rare he ever finds a good man in a position of wealth. When he does, he respects them and they earn his loyalty, even if he's never talked to them. Or at least, what passes for his loyalty.
For a Sheik, he wouldn't take too many risks, or snoop too much. He'd just talk to the other guards, peak in the Sheik's room, do some low level investigation, using his Hat of Disguise to get what info he needs.
So what general feel would he get for the Sheik? Good man? Or in need of some humility?