This 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 (there are 30 Relics in total) which are 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 as some levels drop very useful abilities that let you go much faster (e.g. double jump & sprint).
LEVEL 1: TOAD VILLAGE - 0:17
LEVEL 2: UNDER PRESSURE - 1:16
LEVEL 3: ORIENT EXPRESS - 2:41
LEVEL 4: BONE YARD - 3:44
LEVEL 5: MAKIN’ WAVES - 5:25
LEVEL 6: GEE WIZ - 7:05
LEVEL 7: HANG’ EM HIGH - 8:28
LEVEL 8: HOG RIDE - 9:45
LEVEL 9: TOMB TIME - 10:56
LEVEL 10: MIDNIGHT RUN - 12:26
LEVEL 11: DINO MIGHT! - 13:30
LEVEL 12: DEEP TROUBLE - 15:15
LEVEL 13: HIGH TIME - 16:43
LEVEL 14: ROAD CRASH - 18:10
LEVEL 15: DOUBLE HEADER - 20:18
LEVEL 16: SPHYNXINATOR - 21:43
LEVEL 17: BYE BYE BLIMPS - 23:29
LEVEL 18: TELL NO TALES - 24:33
LEVEL 19: FUTURE FRENZY - 27:01
LEVEL 20: TOMB WADER - 28:57
LEVEL 21: GONE TOMORROW - 30:47
LEVEL 22: ORANGE ASPHALT - 32:36
LEVEL 23: FLAMING PASSION - 34:38
LEVEL 24: MAD BOMBERS - 36:00
LEVEL 25: BUG LITE - 37:42
LEVEL 26: SKI CRAZED - 39:25
LEVEL 27: AREA 51? - 42:00
LEVEL 28: RINGS OF POWER - 44:30
LEVEL 29 (SECRET): EGGIPUS REX - 45:44
LEVEL 30 (SECRET): HOT COCO - 46:41