In reply to TobyA:
Vacuum flasks will lose heat to the outside by three means, conduction, convection and radiation. Let us ignore convection since the gap width between the walls is small, and induction through the plastic cap. That leaves conduction through the walls at the neck, conduction through the residual air in the 'vacuum' space and radiation across that space. If there were no vacuum, but just air at atmospheric pressure, then I find that all three heat loss mechanisms are around 1 W, giving a temperature drop rate of about 2.5 degrees per hour. Does that sound about right? If there were a perfect vacuum then that you would still have a temperature loss rate of 1.5 ish degrees per hour since the radiation loss and conduction loss through the neck will not change. So a real flask would sit somewhere between in this range, and if the vacuum is broken that should result in at worst a doubling of the temperature drop rate.
For the vacuum to make any difference, you need to get the mean free path of the air molecules to be longer than the gap width. Any less than that and the molecules get in each other's way and it doesn't matter how many of them you have. That means that the pressure has to be less than a few hundredths of a millibar, or about 1 Pa, which is one hundred thousand times lower than atmospheric pressure. I don't know if that is achievable in real flasks on the hill.
In working all this out, I used: gap width 5mm, stainless wall thickness 0.5 mm, flask diameter/height 5 cm/20cm, coffee at 80 degrees, outside air at 0 degrees, emissivity of shiny stainless 0.1, 10cm travel length for heat transfer from hot coffee to outside, via the neck, 1 litre coffee, thermal conductivities: air at 1 atmosphere 0.024 W/K m, stainless steel 16 W/K m.