i've experienced the LCD being negatively affected by cold, but usually well below freezing. in this case, it should be OK when it warms back up. i've had this happen with various electronic devices... wristwatches, GPS, etc. i think it's even worse if you somehow get water vapor inside the unit - this can freeze and make it act up.

as to the battery - yes, battery power is greatly reduced by temperature. so if your battery is weak, then it could easily drop to too low power when it's cold.

i used to have the Specialized Pro wireless until i lost it on a major downhill single track - i never had any LCD problems with it in the year i had it - i know i was out in 15-20F at least a number of times for long enough to get it pretty cold.