How accurate are Apps like runkeeper when it comes to calories burned?

It appears to be very inaccurate.

Are you just curious or trying to figure out how to log that?

Let's break down your numbers like this:

Day 2
Pace 1.3 mph
305 calories hour-1
221 calories mile -1

Day 4
Pace 1.5 mph
390 calories hour -1
254 calories mile -1

All things being the same, you would expect these numbers to be pretty much identical. Most calorie calculators just estimate calories burned based on the time or distance of your walk vs your body weight.

What's more interesting is that elevation gain would imply effort. The day you had a greater elevation gain, it says you burned less calories in an hour and less calories in a mile. However, it looks as if the app possibly factored in your pace, as you had a greater pace over the shorter distance. Therefore it likely assumes you more quickly burned calories based on pace even though you likely expended more energy tackling a greater elevation change.

I use my Garmin watch to track my exercise calories. It has a heart rate monitor built into the wrist strap, which I know is less accurate than a chest band. Yesterday I walked for 56 minutes at 207 lbs, and it says I burned 451 calories with an average heart rate of 111 beats minute -1. This number is also likely inaccurate. According to livestrong.com, I probably burned closer to 240 calories per hour. So, my point is that you should take these numbers with a grain of salt.