The water temp changes the CSI (calcite saturation index), which shows the likelihood for corrosion or scaling of the pool surfaces and equipment. Others here could explain the CSI better than me, but that's it in a nutshell.
Plausible temperatures don't have very much effect. If the water is below 70 degrees, you should allow it to warm to room temperature before testing the CYA level. You should always test the FC, CC and PH shortly after getting the water sample while it is still at whatever temperature it started at. With very cold water you will need to mix the TA and CH tests more between drops, or risk missing a slightly delayed color change.