That happens to be the only contest I have never seen. Therefore, I had to go on what the Wiki calls it - which is "The Coldest Adventurer Contest".
I see that the telescope calls it "Coolest Adventurer" - which is to say, the code I wrote calls it that, since the text you see when you look through the telescope obviously doesn't call it anything like that.
I see that adventures.txt calls it Coldest Adventurer Contest.
I see that when we parse the "You have already entered the XXX Adventurer Contest", we look for "Coldest"
To me, that says that the "official" name of the contest is "Coldest Adventurer Contest" and that changing the name that the "telescope" command reports from "Coolest" to "Coldest" is what you are asking for.
Although, since I have never seen this contest, I have no way of testing. (Bale has a theory that your class or path or zodiac sign or something affects the contests. If you don't mind, what class/path/sign were you on when you saw this?)