I would like to speed up the cold start of the GPS module a bit.

I have partially solved this before by powering the GPS module both from the battery and also directly from the USB connector of the flight controller. Before I prepare the equipment for the flight, I first connect the FC to the smartphone with SpeedyBee running and let it catch the 3D fix, the GPS module is powered from the smartphone through the FC and at the same time I see how it searches for satellites. It is nice. It usually finds 6 satellites in two to three minutes, which is just enough time before I prepare the googles, the transmitter and attach battery to the frame. After many intended GPS rescues, I found that a 3D fix with 6 satellites is not exactly accurate, some recommend 8 satellites as a minimum for a good 3D fix. However, finding 8 satellites after a cold start usually takes much longer if the drone is still on the ground and you know, time is always short. The solution is to take off to a height for a while, the number of satellites will grow quite quickly, land and rearm. But this is another waste of time that I would like to avoid.

So my question is, does Speedy Bee support A-GPS? Or is this support planned for the future? If SpeedeBee will provide assisted GPS support for the GPS module through a connected smartphone with a data connection, the 3D fix could be significantly faster and more accurate.

I'm fairly certain that the SpeedyBee app doesn't support assisted GPS. Even if it did, there is nothing in the Betaflight firmware to accept A-GPS data over the USB connection anyway, and inclusion of such a feature is highly unlikely to happen. The Betaflight devs didn't even want to want to implement GPS Rescue feature and only did so reluctantly after lots of pressure from the community, but it is basic GPS functionality at best (i.e. the bare minimum to meet the requirements) and is nothing like the advanced GPS features you get in INAV.

You might have a chance of getting such a feature implemented into INAV if you raise a feature request (or you code it yourself and issue a pull request), but I can't see such a feature ever being implemented into Betaflight.

What GPS module are you using?

HGLRC GPS module with UNX-M8030 chip

And as for the GPS Rescue functionality in the latest version of Betaflight, I have no complaints, it works really well if the 3D fix is on a sufficient number of satellites. In my first LR drone flight, I had a GPS rescue three times after FS and it always worked well.

I had speed up my fix times with a BN-220:
- Move the GPS to a 3.3v pad
- Ensure that the gps cables are twisted and I cover them with kapton tape

If I remember, I let the quad powered from a USB power bank as home or while driving to the flying location.

Regarding the BF GPS rescue, It has save me a couple of times too. But once I almost lost the quad in the ocean as it took a big dive as soon as the GPS rescue was activated. The Flywoo Explorer does not has a barometer and it thought it was 100mts above the ocean, and it was way closer.
After that experience I'm waiting parts to add a barometer, a magnetometer and move to INAV.