This is a new addition to the PS4 Remaster of Crash Bandicoot 2: Cortex Strikes Back that did not exist in the original PS1 version of the game. This new feature becomes available after you have beaten a level for the first time, you basically have to pick up the stopwatch at the levels starting point to begin the timer and you earn a Relic which is either Platinum, Gold, or Sapphire depending on how fast you are. To have any chance of getting the Platinum and Gold Relics you should beat the main story first so you recieve the 'Speed Shoes' upgrade from the final boss which allows you to run a lot faster.
LEVEL 1: TURTLE WOODS - 0:17
LEVEL 2: SNOW GO - 1:36
LEVEL 3: HANG EIGHT - 2:55
LEVEL 4: THE PITS - 4:02
LEVEL 5: CRASH DASH - 5:26
LEVEL 6: SNOW BIZ - 6:55
LEVEL 7: AIR CRASH - 8:24
LEVEL 8: BEAR IT - 9:41
LEVEL 9: CRASH CRUSH - 10:59
LEVEL 10: THE EEL DEAL - 12:43
LEVEL 11: PLANT FOOD - 14:11
LEVEL 12: SEWER OR LATER - 15:24
LEVEL 13: BEAR DOWN - 16:23
LEVEL 14: ROAD TO RUIN - 17:52
LEVEL 15: UN-BEARABLE - 18:59
LEVEL 16: HANGIN’ OUT - 20:50
LEVEL 17: DIGGIN’ IT - 22:26
LEVEL 18: COLD HARD CRASH - 24:31
LEVEL 19: RUINATION - 26:55
LEVEL 20: BEE-HAVING - 28:34
LEVEL 21: PISTON IT AWAY - 30:18
LEVEL 22: ROCK IT - 32:05
LEVEL 23: NIGHT FIGHT - 33:31
LEVEL 24: PACK ATTACK - 34:57
LEVEL 25: SPACED OUT - 36:41
LEVEL 26: TOTALLY BEAR - 38:06
LEVEL 27: TOTALLY FLY - 39:31