They document it in the data sheet, look for a diode with battery backup. You need to keep power to the clock and SRAM for it to remember the position enough to start in a second or two.
To do a low power mode, you need to maintain VBatt above some low voltage - 1.2-1.5 volts or something, and it takes microamps so it isn't hard. Then shut off 3.3v. It won't track but will restart very quickly.
The problem is that on both SFBoBs VBatt is shorted to 3.3v.
(To Sparkfun - instead of a solder jumper shorted by default, you could put a schottky diode across the two - how expensive is the one they recommend in the data sheet? And do you know how hard it is to find an SMD diode to fit the shorting jumper?).
And Why it might take so long to start:
It needs power to remember the time and position for a fast fix. It doesn't say this but it may take several minutes under a clear sky from a cold start since it has to download all the data from several visible satellites before doing anything.
In "low power mode" it uses two search engines, in enhanced it uses four. If it has no idea of where and when it is, it needs to scan the entire GPS band (remembering doppler) until it hears something, tune and compensate, and when it tunes the signal sufficiently search for another sat while another section downloads the ephemeris, all which can take a while (and it needs a lock - a good signal - to do the download). That is why there have "61 channel" when only at most 14 might ever be used - to scan and do a first fix much faster, i.e. with 60 channels all looking they can each search 1/60th of the band.
This is what AGPS will bypass and why it also needs to know where you are in some cases. The AGPS download is FTP so it might be a problem depending on firewalling or other settings.