The thing with specifications is that when used within the specs the part is guaranteed to behave as specified, and when used outside the specs, the part is not guaranteed to work at all.
Surely there is a margin for manufacturing tolerance and defects that allow the parts to be binned for narrower or larger temperature range.
And each manufacturer may manufacture the parts in their own specific way so there is no general answer how your specific part will behave when you bring it to lower temperatures than specified.
It depends on the manufacturer, manufacturing tolerances and which unique properties the parts you bought happens to have.
You might be able to guess how it will behave, if you open up the manufacturer datasheet for the part and seek any parameters or curves that are listed at different temperatures.
To guess what might happen, i.e. what can be out of specified tolerances :
- Output voltage may be higher
- Output voltage may be lower
- Output load regulation may be worse
- Output thermal regulation may be worse
- Adjust pin current may be vary more
- Output max current may be more or less
- Output short circuit current may be more or less
- Ripple rejection can be worse
- Dropout voltage may be more or less
Bear in mind that temperature does not only affect the regulator, but also nearby parts such as capacitors, so when the temperature changes, so does the capacitance and effective series resistance of capacitors, so as a whole the system may just be a poor and unstable power supply at low temperatures.