An LM317 configured as a constant voltage source depends on the ratio of two resistance values (R2/R1), so the *tracking *TC (i.e. how well the TCs between the two resistors “track” over temperature) is more important than the absolute TC of the resistors. Therefore, the resistors should come from the same lot at the very least. Ideally he would want to use a monolithic resistor array containing both resistors, but a COTS array contain the values he needs might be hard to find. The most practical approach is to use resistors manufactured from the same lot, and then use a thermally-conductive potting compound to keep them at the same temperature.
The most serious problem when operating electronics at low temperatures is not parameter drift of components - it’s the condensation of water. This is generally solved through the use of conformal coatings. But applying a conformal coating to a PCB is not as easy as it might seem; it’s not a matter of simply spraying something on the board. If you don’t do it right, you will end up with a coating that does a good job of trapping moisture, which means you would have been better off not coating it at all.
You people need to get out more often.
This is not a Military grade application.
Even the absolute cheapest carbon-film resistor that Digi-key sells has a worst-case TC of 350ppm/C. That means that if we assume a ±50C temperature range around room temperature, then if one resistor increases in value by 350ppm, and the other one decreases by 350ppm, the overall change in value of the divider is only 700ppm X 50 = 3.5%, which is well within any power supply spec for a consumer product. And, this is by far the worst case.
Guilty as charged. As an EE who works on military stuff, I have a hard time being loose with tolerances, even when the application calls for it. Am always thinking about third and fourth-order effects, and how they might make a missile veer off course. 
Well, I’m proud to have you on our side…
Second that last quoted comment! 
Come on, people. I’m slapping these components to a piece of wood, carved to fit in the battery compartment of the GPS, and maybe slapping some silicone caulk over the thing to keep it dry! The voltage has to be something like 3v, ± 0.5v. And I don’t care if it stops delivering voltage if it gets too cold, I only wanted to know if it might exceed 3.5v, possibly frying the GPS if it got too cold. I’m not building a military grade missile, here, nor a space-rated Jupiter probe. It’s just a freakin’ car power supply for a normally AA battery powered device. I only wanted to know if the LM317T would not fry my GPS if it got too cold!
Let’s get some perspective, here! 
[Walking timidly into the room]You’ll be O.K… design it for 3.0 V, and make sure to use a couple fixed resistors (as opposed to a fixed resistor and a pot). You can also try adding a 3.3 V or 3.6 V zener diode across the output to help protect the unit. If you don’t have a 3.3 V or 3.6 V zener diode handy, and still want some over-voltage protection, I suppose you could string six silicon diodes in series and hang the string on the output. [/Exiting timidly from the room]
Hey! I like that zener diode idea. I just might do that! Thanks!
Does the thing have a crystal controlled clock in it for any reason? I have no idea if it does or does not. If it does, temperature extremes might adversely affect the frequency; if this is not a problem, ignore me. Everyone else does.
Gee, I hesitate to suggest alternatives after all this, but what kind of link cable does your GPS use? I have a couple of handheld receivers with USB link cables and I can just run them off the car by plugging the link cables into one of those cheap iPod car chargers. On one of them you have to dismiss the link menu that comes up automatically, but they both will run indefinitely with full backlight when plugged in.
There are no reasons for voltage regulators to be clocked.
Coming back to this later, all I have to say is "hey Crafter_Man, any time you want to “whisltle into my room”, timidly or otherwise, you are welcome. Once you get the idea of what I’m trying to do down straight, your suggestions are potential gold! Your “zener diode” suggestion gets the “silver medal” for “best idea”, after the “put it in the freezer and measure it” idea. The only reason you missed the gold, is I’d have to actually buy a zener, and I’m trying to do it with scrap. Seriously, dude. I’m glad you are on “our” side when it comes to designing missiles.
ETA: I see a couple more responses since I read the last and started composing this. I’ll get back to you immediately. Edit window too short.
OK, LouisB? Leaffan addressed it. No further comment from me.
GreasyJack, the manufacturer doesn’t make or sell them, anymore, and even when they did (my brother has one, too, and also has the cable), the cable didn’t supply power from the computer to the GPS. It only transferred data (by a standard I’m not clear on, and my brother ain’t going to let me use his for potentially device-frying experiments with), so that’s out as a potential solution.
Will a big-boobed nympho be in the room, too? If so, I’m game.
If I could find one of those, do you really think I would be playing with electronics? :rolleyes: