My understanding is that this was actually intentional. In the northern parts of Scandinavia, farmers allow their herds to free range on farms that are frequently bisected by roads. During the wintertime, due to the hazardous conditions, it's not uncommon for careless motorists to hit their reindeer by accident. As a sort of warning, some farmers will put the frozen corpses up as a reminder to motorists that they are entering an area where they may encounter their reindeer. The corpse would have been put up "whole" and periodically scavenged by carrion birds or wolves during warmer days...leaving examples like this.

I'm not 100% on this and may be entirely incorrect, however it stands as a better explanation than it froze to death standing up. Reindeer have survived for 1000s of years in this environment. It's unlikely it would have frozen to death. In the unlikely event that it did though, it'd most certainly do so on the ground after slowly becoming weaker and weaker.