A Lambert quadrilateral is a quadrilateral three of whose angles are right angles. And in 2-d hyperbolic space $\mathbb H^2$, we have nice formulas for the fourth angle.

If $AOBF$ is a Lambert quadrilateral in $\mathbb H^2$, with $\angle A=\angle O=\angle B=\pi/2$, then $$\cos\angle F=\sinh |OA|\cdot \sinh |OB|.$$ My question is, do we have a some kind of comparison result for $\angle F$ of Lambert quadrilaterals with the lengths $|OA|$ and $|OB|$ fixed? (maybe as corollary of some classical comparison theorems?)

By this I mean that if we consider another Lambert quadrilateral $A'O'B'F'$ with $|O'A'|=|OA|$ and $|O'B'|=|OB|$ on a compact surface $(M^2,g)$ with curvature $\kappa<-1$, can we compare the two acute angles $\angle F$ and $\angle F'$? In this case it seems natural to expect $\angle F'\le \angle F$.