Quite simple really...honestly ice temperature depends very little on ambient temperature. Underneath the ice (which is about anywhere from 1 1/2" to 1 1/4" thick) is a concrete slab. Encased inside the slab is a grid network of either copper or steel tubing which carries a brine mixture chilled to around 8 or 10 degrees F. This network is fed by a refrigeration system that can basically be thought of as a giant air conditioner. Without getting deep into the workings of refrigeration all you really need to know is that the brine is pumped through a "chiller" which is filled with R22 refrigerant kept at anywhere from -22 to -25 degrees F. When the brine comes out it is around 8 or 10 degrees as I said before. Once it gets pumped out to the floor it comes back anywhere from 5 to 10 degrees warmer since it is removing heat from the floor. Then of course the cycle repeats itself. As a result, the floor is extremely cold and surface temperatures on the top of the ice can range anywhere from 18 to 19 degrees F.
We use hot water (very hot-160 degrees F is optimal) in the Zamboni because, weird as it sounds, hot water freezes faster than cold water. Since the water is already giving up heat at a rapid rate in the form of steam, once it is spread out on a very cold surface it freezes quickly. An added plus is the fact that initialy it melts a very small part of the top layer so instead of just freezing on top the ice it "meshes" with the bottom layer.
Anyway, getting back to the question of the air temperature inside the arena, the most important factor is actually humidity. If you could measure it you'd find that it was very low compared to outside. With a full house (people exhaling produce A LOT of water vapor) the humidity might range anywhere from 30-40%. Humid air transfers heat much faster than dry air, so if the air was warm AND humid there would be problems. Believe it or not there are many outdoor rinks out west that keep ice through the winter despite temperatures of 60-70 degrees because the air is so dry. Many people don't know that the Zamboni was invented in Paramount, California at a rink that was owned by the Zamboni family. Although there is a second manufacturing plant in Ontario, Canada the headquarters remains in Paramount. You can read more about it
here.
Kind of long winded I know, but it's useless knowledge I love to use once in a while
I will be there around 9, for anybody who is interested in sharing a beer. 
