In Temple Run, you run away from stuff. As you run, popups come up every 500m to tell you your distance. As you get farther, you seem to run faster. Here's the question: do you actually run faster? I mean, do you cover more distance, allowing for a shorter time interval between 500m markers? Or do the 500m markers just show up every x seconds, and the game speeds up as I get farther to increase difficulty?
Using the timestamps, I timed this video at about 250m every 10 seconds.
That video only goes to 1000m several times though, so I also stopwatched my own run up to 2000m using fast app switching (so a slight inaccuracy involved). Up to 2077 m I came up with 22.33 m/s, which is a little less than 250m per 10s, just like observed in the YouTube video. Based on this I conclude that you do not actually accumulate distance traveled faster, even though you appear to be running faster. The apparent increase in speed seems only there to increase difficulty.