Your max HR is the same for each mode of exercise when you reach that same HR for given exercise. Example, if you reach a HR of 120 BPM cycling and running it doesn’t change.
As far as calories expended, this is an intriguing thought. Let’s think about that. Calories expended are not only based on O2 consumption, they are also dependent upon the muscles involved. Calories expended are also based on age, sex, height, and weight. For the sake of this answer, we’re talking one person, not many. So, for this one person doing two different exercises, the following would apply.
Calories are energy. It takes more energy to move more muscles. Cycling uses less muscles than running. Running on a treadmill uses less muscles than running outside. Through this thought process, it stands to reason that you would need to work harder on a bike to reach 120 BPM versus running outside. Since using more muscles burns more calories, it stands to reason that 20 minutes of running at 120 BPM would use more calories than 20 minutes of cycling at the same HR. Unless the O2 consumption is the same. More muscle involved means more O2 is required for the steady-state cardio (aerobic). One could also say that the HR is related to the body’s O2 needs. In other words, 120 BPM; regardless of muscles involved; are going to use the same amount of calories no matter the modality of exercise. The only real way to tell is to use a metabolic cart.
Let’s not forget about excess post-exercise oxygen consumption (EPOC). The more muscles involved in an exercise (aerobic or anaerobic) the longer the recovery from that exercise, which means more post-oxygen consumption. The more EPOC needed, the more calories you burn.
In my opinion, you would burn more calories running outside 20 minutes than cycling 20 minutes at the same heart rate. Because running uses more muscles, which requires more energy (calories) during the exercise as well as post-exercise during recovery.
The EPOC is one variable heart rate monitors and treadmills can’t measure.
Good question. Made me think.