There have been many claims that exposure to high altitude increases energy expenditure. While overall this is likely the case, I was interested in seeing if there was information specifically addressing 1) Is there an increase in basal metabolic rate (BMR) alone or 2) Is there also an increase in energy expenditure during exercise and 3) Are any increases actually due to the increased hypoxia, or is it something else?
I looked up on PubMed and found a 2006 review paper that investigated weight loss at altitude. Unfortunately the abstract says little - I have the actual article but I am not "allowed" to provide it (that whole copyright thing) - you can contact me for more detail if interested.
The paper in general shows that there hasn't been a whole lot of controlled studies to answer these questions, but do have some tidbits
1) One study (Nair 1971) had 2 groups with group A exposed to hypoxia alone for 3 weeks then cold & hypoxia. Group B had the reverse order. Under hypoxia alone, group A actually had a decrease in BMR after 3 weeks. After inclusion of cold temperatures, BMR increased. In group B, exposure to both hypoxia and cold increased BMR after 3 weeks. After taking away the cold, BMR stayed elevated.
The results are definitive but point to cold temperatures being the significant factor in increased BMR at altitude. Another study showed increased BMR after exposure at 4300 m for 21 days, but did not distinguish exactly what was causing the increased BMR. The review article states there is no conclusive evidence and more research is needed to see if hypoxia affects BMR.
So that address #1 and #3 (somewhat), what about #2? Well there seems to be even less controlled work done on this. The article states (and what I have believed) that work requires the same amount of oxygen at high altitudes as at sea level. However, the maximal work output (VO2max) is reduced and thus all levels of exertion are more tiring. But there doesn't seem to be any studies really showing comparative energy expenditure at altitude versus sea level, at least according to this review.
This goes back to the idea of eating high carbohydrate diet at altitude. You'll have an increased BMR (at least due to cold temperature exposure) but more importantly you'll burn relatively more glycogen at a given workload than you would at sea level. This goes back to how our aerobic energy pathways utilize fats and carbohydrates depending on intensity - the more intense, the more glycogen (carb) is used instead of fatty acids.
Ascending from 13,000 ft to 14,000 ft may not be more work than 1,000 ft to 2,000 ft, but the lack of oxygen forces the body to get more energy out of that oxygen, and therefore uses glycogen reserves more (both aerobically and anaerobically) than fat.
The article goes on to talk much about appetite suppression as being a larger factor in overall weightless at altitude.
So overall, estimation of caloric expenditure of a hike may not be affected by altitude - but a person's basal metabolic rate will likely increase causing an overall increase in expenditure. Even at the same caloric expenditure, glycogen reserves are used more and therefore a high carb diet is beneficial.
There may be pertinent research that this review article didn't address, so if you find some it would be great if you let us know.