I've created a variant of the breakout board that has a super capacitor (100 mF) to provide about an hour of battery backup. This allows the module to warm-start very quickly across brief power outages.

And, sure enough, it warm-starts almost immediately when backup power is available.

But the module I'm using is the timing variant. While the backup power does preserve the time and almanac so the first-fix is quite fast, the module does not preserve the surveyed position. That is, the module will immediately restart the 2000 second position survey.

That's not as bad as it sounds. The module does provide good PPS fidelity while it's performing the survey (as long as it has good reception), but for the duration of the survey it won't provide the advantages that come from static mode - the ability to give good timing solutions with as few as a single satellite.

What that tells me is that for a GPSDO, the backup power provisioning is unnecessary. Even when completely cold-starting, the module can get a fix in around 30 seconds or so. For an OCXO based GPSDO, the oscillator is going to spend perhaps the first hour retracing anyway, so that extra 30 seconds really doesn't matter.

That said, if you want fast startup times, the supercap will do it (as long as the power outages are an hour-ish or less).