Deer will do just about anything to avoid getting eaten. Their feet might get cold. They might know their feet are cold. But it is highly unlikely they are wondering to themselves how much better their lives would be if they were hanging out at the beach. Because all they know is "survive today", their perspective is a little different.
What I meant to say was yes, deer will walk through freezing cold water, and thin ice, if necessary. Hunting pressure, terrain, other predators, a hot doe, many things determine if it is necessary.