Display may get sluggish and/or dim to the point where it's unreadable, but I've never had a bike comp stop logging due to cold. The information is all there and visible, once the computer warms up again.

Weak batteries may cause the computer to stop logging data completely, and as mentioned, some battery types are more prone to freezing than others. This is why I downgraded from wireless back to wired computers at one point. My bike stays out during the winter, so I was constantly changing batteries for the transmitter unit. There's less hassle with wired computers in that regard.

--J