I've scoured the PHB, DMG and MM and can say with some confidence that the RAW in the currently-published core rulebooks don't mention this point. This is part of the deliberate design of 5e, where not all details are to be covered in official rules.
The DMG gets tantalizingly close in its rules for flying mounts on DMG 119, but that's not your question.
The designers of 5e D&D have deliberately not dotted every i and crossed every t, and leave such points precisely to DM discretion.
This design philosophy is well-expressed on the WotC website: http://dnd.wizards.com/articles/features/philosophy-behind-rules-and-rulings
The DM is key. Many unexpected things can happen in a D&D campaign, and no set of rules could reasonably account for every contingency. If the rules tried to do so, the game would become unplayable. An alternative would be for the rules to severely limit what characters can do, which would be counter to the open-endedness of D&D. The direction we chose for the current edition was to lay a foundation of rules that a DM could build on, and we embraced the DM’s role as the bridge between the things the rules address and the things they don’t.
It sounds like you know what the rules do say (rules for mounts, rules for PCs pushing themselves past 8 hours, the exhausted condition, rules for flying mounts). It is your job as DM to form a bridge between these rules and the specific situation which has come up in your game of letting horses rest between gallops.
And if there is any doubt about this applying to the case in question, here is a quote from Mike Mearls in an interview posted on Reddit:
This [travel pace] is an area where we erred on the side of ease of use. Someone (maybe Crawford?) did some research on travel times of mounts vs. people on foot, and came to the conclusion that it was pretty exceptional and required a lot of support for mounts to provide a significant, long-term speed boost.IIRC, the Pony Express required riders to swap out horses several times per day, and the animals themselves had serious risks of injury.So, for that reason and to keep things simple, we standardized travel times.
This quote shows that, as far as the D&D lead designer was concerned, the level of detail for overland travel was set low, and presumably only the rules considered essential were included in the published rulebooks.