Due to the long duration on Hex, it is clearly intended to function across multiple encounters as described in this answer.
Due to the extended duration, it seems feasible that a warlock could take a short rest to recover their spell slot and thus go into a future combat with a Hex at the ready and a full assortment of spell slots. Is this correct? RAW do not appear to prohibit this.
Concentration is ended by (PH chapter 10):
Casting another spell that requires concentration.
Taking damage.
Being incapacitated or killed.
Or "The DM might also decide that certain environmental phenomena..."
As a short rest does not require casting any spells, cause damage, inflict the incapacitated condition, kill you, or cause any environmental phenomena, it does not end concentration.
The only things that break concentration are listed in the PHB, the PHB says on 203-204:
Normal activity, such as moving and attacking, doesn't interfere with concentration. The following factors can break concentration:
Casting another spell that requires concentration. ...
Taking damage. ...
Being incapacitated or killed. ...
Unless you fall asleep, you are able to maintain concentration on the spell. There has been some question on forums about the reverse, that is "Can you really rest if you are concentrating on a spell?"
The answer to this question seems to be yes also, as:
[S]hort rest is a period of downtime, at least 1 hour long, during which a character does nothing more strenuous than eating, drinking, reading, and tending to wounds.
I'd argue reading would require a similar amount of focus as concentrating on a spell. The designers never commented on this aspect, but their posts about Hex make me believe you can indeed carry it through a rest, and benefit from the rest.
And Mearls answers this question on twitter, which indicates it is the developer's intention, even if it isn't an official RAW ruling:
Q: Can I start a short rest while keeping concentration on a spell like Hex?
A: as long as you don't sleep
The definition of a short rest from the rules is as follows:
[A] short rest is a period of downtime, at least 1 hour long, during which a character does nothing more strenuous than eating, drinking, reading, and tending to wounds.
Concentrating on a spell is clearly not an activity which is trivial but rather requires mental effort; otherwise there would be no justification for the single spell limit and the term 'concentration' would make no sense. It implies that the character requires continual effort to maintain a spell and this breaks the 'nothing more strenuous' clause. A character therefore cannot take a short rest while concentrating on a spell.
So a character could maintain their spell whilst their companions took a short rest, but they could not gain the benefits themselves whilst maintaining concentration.
