Does elevated heart rate burn more calories?

I think riverside's answer is the right one. I also think the numerical difference - if there is one - kind of gets lost in the overall noise of food/activity estimating errors, i.e., it's small, not a big worry.

I also think, if it came down to precision (which is unrealistic ), a factor in whether 1 minute of rope-jumping calories generalizes to 60 minutes of rope jumping calories just by multiplying by 60, is whether as a practical matter you can sustain the same intensity for the duration. I'm not crazy-unfit, and the maxiumum intensity I can sustain for 1 or 2 or 5 minutes is a way higher intensity than I can sustain for 60 minutes.

I can't prove it scientifically, but I'd bet that the calorie burn for jumping rope pretty much depends on how many calories are involved in one jump of X height with Y rope-turning energy, multiplied by number of jumps - that's the work, in physics terms. A person might jump pretty fast/high/energetically for one minute, but I'm thinking the intensity drops over the course of that hour, and with it, the per-minute calorie burn . . . no matter what the HRM says because it's deceived by how HR keeps drifting higher.

Speculating, though.

For sure, the heart rate drifting upward doesn't equate to higher calorie burn in and of itself. If it did, we could burn calories by watching scary movies, while sitting completely still.

Since heybales' post (good, as always) suggests we can offer n=1 case study reports, here's one, from grandma, FWIW, of a slightly different nature:

Yesterday, Friday, I did an ultra-slow rowing machine workout, quite short. I often do repetitions of 2000 meters.

Both yesterday and the day before (Thursday), coincidentally, I did pretty much the same workout, but at much different intensities. It was 2 x 2000 meters, with a short break (2 minutes) between. That break was a combination of easier rowing & drinking some water while not rowing, both days. Both days, I did a 2-minute very easy cool-down at the end. Thursday's workout was more than averagely intense for me, Friday's was much less than averagely intense (barely even aerobic ), because reasons.

This rowing machine (Concept 2) has a good power-metering system (measures watts of power the person produces). Watts (for the same person at the same reasonable efficiency, corrected for body weight) translate reasonably accurately - and very consistently - to calories.

It's a reasonable assumption that my fitness level, body weight, and efficiency are pretty similar, on 2 consecutive days (without any huge lifestyle perturbations to cause changes in between). So, I expect consistency in the calorie estimates both days. I'm not an awful terrible rower (according to qualified coaches ), so I think my efficiency at directing work into the flywheel is OK-ish, thus I'd also expect reasonably accurate calorie estimates from the machine.

According to the machine, adjusted for body weight, Thursday's fairly intense 2 x 2k burned 191 calories for just the 2k parts. (It took 20:13.6 because like I said, I'm grandma.) According to the machine, again adjusted for body weight, Friday's completely easy 2 x 2K (22:00.9) burned 180 calories. Pretty darned close to the same, which makes sense because it was close to the same number of strokes (20 spm average, both days, but a wee bit more time, Friday).

According to my Garmin (with HRM), Thursday's row (the more intense 2k portions plus the easy rowing during the break, plus the cool-down) burned 172 calories (clock time, 24:19). Average heart rate, 140, max heart rate, 157. (My age-estimated HRmax is 156. Actual max is around 180, and Garmin knows this.) Friday's easy row, Garmin thinks, burned 146 calories (clock time, 26:09). Average heart rate 121 ( ), max 134.

Admittedly, the HRM data includes the break and cool-down and the rowing machine estimate doesn't, but those parts were 471 meters Thursday, and 496 Friday, so probably not a big caloric difference there between the days. Extrapolating, those extra 470m+ parts ought to be worth around another 21 calories on top of the rowing machine estimate.

So, machine sees 11 calories difference between the days, which makes sense to me, because it's about the same amount of work (covering 4K total) on each day. HRM sees almost 3 times as big a difference in calories (26 calories). HRM gives total estimates a little lower than the machine, but that estimate includes more work: The rest interval rowing & cool-down, but only 25 meters difference in that between the two days - less than 3 typical strokes' worth.

Based on all of the above, my best inference is that my HRM underestimates my (gross) rowing calories quite a bit more dramatically for a slow, easy row as compared with a harder one. This is without even considering that the machine's calorie estimate didn't include the almost 500m of rest/CD rowing.

Also, none of those calorie-estimate differences are worth worrying about. It's like being a few grams off on weighing the olive oil I put in my sauteed zucchini at dinner. Ditto a long meandering grocery trip at the big-box store vs. running into the convenience store for a carton of milk, or similar NEAT variability. NBD.

Any of those exercise estimates would be workable, for practical calorie counting use. 191 vs. 146, with a TDEE around 2000 +/-, is not any kind of big deal. (The difference between gross and net workout calories, which almost no one worries about, is numerically about as big a deal, maybe bigger.)

FWIW, the last (short) lifting workout I recorded on my Garmin (for rep counts, not calories) estimated at 74 calories, with average HR of 104, max of 126. MFP, whose METS-based estimate I actually logged - tiny though it was - was 57 calories. It's possible that this is a tiny underestimate, because I'm super impatient, so the only way I can make myself lift is to (dysfunctionally ) superset everything and take essentially no rests. My heart rate doesn't spike during lifting, generally, much beyond levels I see walking around doing daily stuff. Anything about these differences, too, is small potatoes, for me. I use the METS estimate anyway.

Small numbers. They average out. We monitor our loss rate, and adjust our calorie intake. It all works out. It's good to understand where major errors could creep in (like strength work if it spikes HR, or HIIT's HR recovery/drift issues), but most of the time, fitness tracker data will work fine, as a practical matter.