If you want the best longevity and highest stability, it makes sense to leave a GPSDO on all the time. Also, where you place it matters too, try to put it somewhere where the ambient temperature doesn't vary a lot. That means don't put it (as I did with mine at first), by a window next to a hydronic heater/radiator and outside wall.
Attaching it to an inner wall might work out okay. An inner closet away from exterior walls might be the best place for it in a typical home. Its more likely to be kept in a fairly comfortable temperature range without wild swings in some kind of enclosed insulative container within the more occupied areas of a house.
Note- I am not going to speculate about equipment longevity below, i am just going to speculate on accuracy of the frequency and time measurement aspect of GPSDOs versus other sources of time.
With the caveat that 1. Other people here know a lot more about this than I do. and 2. My own equipment isn't this good so I am just repeating what Ive read..
its my understanding that the following usually applies but only up to a point..
Not as much when your application starts to get into the range where nanoseconds matter..
A GPSDO is a sort of intermediate standard that excels at some kinds of things and isnt the best at others.
Instruments that use an external GPSDO should usually rely on that.. assuming its a good GPSDO, its likely it's stability is superior to most other oscillators of the kinds most people have access to, (unless they are "time nuts" or calibration labs..) Also, its verifiable.. its capable of being NIST certified.. which may or may not matter..
So then, in most contextx it would usually become immaterial how stable their own reference oscillators were, if it was pretty clear that although good they were unlikely to be as good as a GPSDO.
unless that oscillator for some reason was preferable to the GPSDO A free running oscillator may be more stable than a GPSDO at a very fine level of detail, basically.
In some critical applications where the free running oscillator is very good, the GPSDO might not be as stable when taking measurements that relied on a short time constant.
A rubidiium frequency standard or perhaps a very very good OCXO.. with exceptionally low jitter might be better, substantially better for those things than an average quality GPSDO.
NIST has free publications that do an excellent job of explaining this.