Knack 2 has 5 Time Attack Challenges. Getting 5 stars in all time trials unlocks the "Speed Demon" trophy.

Time Attack is unlocked after beating the story for the first time. It can then be found in the main menu under "Extra Content". The faster you are, the more stars you will earn. 5 stars is the best score.
Make sure you kill all enemies along the way. Doing so freezes the time for a few seconds. The green rings freeze time, too! These challenges are quite doable with a little bit of practice.

If you have trouble, unlock the crystal knacks first. I highly recommend the Speedy Knack (Aqua Blue) and Saber Knack (Topaz Yellow).
Speedy Knack runs and attacks a bit faster. Saber Knack can one-hit most enemies which saves a lot of time during combat. You unlock the crystal knacks by opening treasure chests.

TIMELINE:

Time Attack #1 - 0:05
Time Attack #2 - 1:09
Time Attack #3 - 2:15
Time Attack #4 - 3:21
Time Attack #5 - 4:38