Super Mario Galaxy/Freezeflame Galaxy

Freezeflame Galaxy is accessible from the Bedroom observation dome. You need 26 Stars to enter.

Star 1: The Frozen Peak of Baron Brrr
Boss fight.

Star 4 (Cosmic Comet): Frosty Cosmic Mario Race
This Star is available when a Prankster Comet is in this Galaxy.

Really easy, unlike the other Cosmic Comet races. Simply run across the ice, skate when it's not too dangerous, and get to the star before your double.

Star 5 (Purple Comet): Purple Coins on the Summit
This Star is available when a Purple Comet is in this Galaxy.

You need to collect all 100 Purple Coins, with no time limit.

There are quite a few hidden coins, like two hidden by the base of the slide at the beginning of the level. Make sure to look carefully for any not-so-obvious coins by changing the camera whenever you can. If you miss a few coins, it's okay because there is a Sling Star close to the summit that you can use to get back to the bottom.

You may have trouble with the higher coin of the two on platforms beside the fountain at the base of the level. The easiest way to get it seems to be to grab the Ice Flower, run to the platform (making sure not to skate) opposite of the one with the high coin, and make sure that Mario is hugging it so that he doesn't slide. Then, run to the fountain and jump on it, do a double jump from the fountain to the platform, and finally do a triple jump and a spin to get the coin. Remember, this has to be done quickly because the Ice Flower might run out on you, so if you mess up with a jump, don't take a risk. Just go back to the platform with the Ice Flower, and wait for it to reappear.

Star 6 (Hidden): Conquering the Summit
Enter Star 1.