It's nothing to do with temperature, it'll just be that it's acquiring a fix each time and it gets more accurate as the car warms up. One doesn't cause the other!
Indeed - post hoc ergo propter hoc
This is a feature of the way GPS works. The satellites aren't geostationary but rather in low earth orbit so they're all constantly moving relative to the surface of the earth. Once a GPS received has acquired a few satellites, it will continually track them, adding new ones as they come into view and losing some as they disappear from view.
When a receiver is switched off for a time then, when it's powered back up, it won't know where the satellites are. It can remember it's own last location and the position of the satellites as much as it likes, the problem is the satellites will all have moved. The longer the receiver is shut down, the further the satellites will have moved.
In this situation the receiver has to do a cold start and acquire the satellites from scratch. This can be a time-consuming process so there are various ways it can be expedited. One of the most common is called AGPS (A for Assisted), which is what smartphones use. This uses an Internet connection to obtain the current position of the satellite network from a ground-based server, so it doesn't have to hunt for them itself.
Obviously a car won't have this luxury (even with Connect I doubt it would do so) so they use other tricks, such as powering up the GPS receiver the moment the car is unlocked to give it as much of a headstart as possible. Also, OEM systems built into cars are often better at finding satellites quickly than standalone battery powered units, partly because they have a nice big roof-mounted aerial to work with and also because they have more power available. GPS receivers are a notorious power hog (anyone who uses smartphone nav will tell you this) and the more juice you have available the easier it is to detect satellites.
