Was any in-universe explanation given as to why Destiny couldn't handle leaving and re-entering FTL quickly? In Blockade they said the (minimum) delay was 3 hours, and Rush was concerned about jumping the gun by as little as 15 seconds.
Short Answer: The Destiny's FTL drive system had been compromised and one of its 16 drives had been damaged and unable to be replaced. Rush presumed that any use of the system without proper calibration and without allowing the 3 hour recovery time could endanger the systems causing permanent damage.
Longer Answer: Destiny has an FTL drive system. Destiny was originally equipped with sixteen FTL drives but one was damaged and could not be replaced. There was a question of whether it was possible to engage the FTL with fewer than fifteen drives. The FTL drives also required the ship's shield system to have at least 5% power to protect the ship until the FTL drive is completely active.
The Stargate Wikia confirms:
The Destiny's FTL system must remain active for four hours after a jump and inactive for three hours when they are disengaged. Jumps made earlier than this can cause damage to the engine.
The Faster-Than-Light engine, or FTL engine, is a technology used on the Ancient ship Destiny and the Seed ships sent ahead of it. It is capable of faster-than-light travel without entering hyperspace. FTL has also become a Tau'ri expression for an engine capable of travel at speeds greater than that of light, whether it is through use of hyperspace or not.