The problem domain
You have a shelter that you use in a cold climate, and you want to provide it with some heat to take the edge off the chill.
You already have a structure in a location which gets cold winters, let's say in the -15C to -30C range. When it gets down to -30C you are used to turning on a heater, whose fire usually increases the temperature by approximately 10 degrees (to -20C).
You want to construct a similar structure - exactly like the first, or as close as possible - except this one will be in an even colder area near one of the planetary poles and getting winters that might reach temperatures as low as -80C. Similar to before, you plan to turn on the heater if you wake up to the lowest expected temperature of -80C.
I want to figure out how much the heater will warm the structure in the colder area. It very well may just heat the structure by 10 degrees as before up to -70C, but I do not know for sure.
This question does not seem to be as straightforward as I originally thought it would be, and there have been other comments and posts probing some specifics here and not getting me to a satisfactory answer, so the question is this...
What are all the non-negligible forces at play here which should be considered to decide if the secondary structure will be heated by the same amount (10 degrees) as the first structure, and in what way does each of these affect the result?
In case it matters: The building will likely be either a cold-weather tent or a human-made cave made from snow which is designed with the floor higher than the doorway to trap the warmer air inside. If the specific choice matters, feel free to either say so in comment or assume one or the other for your answer.
I think the primary concern might be the
heat capacity (ie: the number of heat units needed to raise the temperature of a body by one degree) of air. See also my question Physics.SE: If two objects of different temperatures have the same heat source applied, do they heat up by the same amount? for the Q&A which led me to this (hopefully correct) assumption. One of the answers even seemed to suggest that the heat capacity change may be significant, assuming I read the complicated answer correctly (maybe not).
I tried to Google up whether the heat capacity of air changes with temperature. I found an interesting table labeled "Specific Heat Capacities of Air", though that does not go down far enough, and a calculator which does. I get 1.005 kJ/(kgK) for 0C and 1.009 kJ/(kgK) for -80C. In my naivety, I would guess that means the heat source might work 0.4% better at the colder temperature (making this tree I'm barking up a negligible one), but I would not be surprised if I am way off base.