Physics tells us that it should take the same amount of energy to move a given weight (our body) over a given distance regardless of the air temperature. In organisms that need to maintain a steady temperature (like us), physiological processes may come into play that alter that equation somewhat.
When we exercise, our muscles give off some of their energy as heat. We warm up, becoming not only more flexible but also hotter. We may need to shed a layer of clothing as our workout progresses, but that may not be enough.
The American Council on Exercise (http://www.acefitness.org) explains that in hot weather, our body has to make some changes to get rid of its excess heat. Increased blood flow to our skin triggers sweating, which leads to evaporative cooling. Our heart has to beat harder to move the blood out. In cold weather, heat is radiated from the body without the need for sweating, so in theory warm or hot weather exercise would burn more calories. This would hold true unless we become so cold that we start to shiver, which would increase our calorie output beyond the exercise itself.
In practice it’s a little more complicated. A study of yoga students who had classes in either hot or normal-temperature studios, cited in http://www.livestrong.com, found that the students in the hot class thought they were working harder than the students in the normal temperature, but physiological measurements showed that the energy output between both groups was about the same. Apparently the sweat and breathlessness brought on by the heat tricked the hot yoga students into thinking they were exerting themselves more than they really were. Because the body doesn’t want to overheat, people may even back off a bit without realizing it.
In an article at http://www.active.com/running/, exercise physiologist Jack Daniels explains that the number of calories a runner expends per hour depends on the runner’s speed and how efficiently the runner uses oxygen. Both can be affected by temperature but not unless the temperature causes the runner to change his or her pace.
For example, a runner on a snowy day may be slowed by bad footing or bulky clothing. A runner on a summer day may have to check his or her pace to avoid overheating. The calorie output per mile would remain the same, but at a lower speed, it would take longer to burn those calories.
A study of cyclists working out at different temperatures, also cited by the American Council on Exercise, found that they burned more carbohydrates at lower temperatures, more fat at higher temperatures, but lasted the longest at moderate temperatures, around 50 degrees.
So moderate temperatures are probably best for outdoor exercise, but hot, cold, or in between, the most important thing is to get out there and do it!