The Hat Kid's Diary records the Hat Kid's adventures from the start of the game onward. It can be found in the secret hideout under the pillow pool in the Bedroom, which requires the Ice Hat to reach. Once the Ice Hat is acquired, there will be a yellow column of light revealing the location. The entries are labeled by the day(e.g. Day 1, Day 2, etc...), and there is one for each act, but they are not necessarily dated in the order of the acts. Some entries cannot be read without cheating as their levels must be played before the player can get the Ice Hat. The diary does not affect game-play, but there is an achievement for finding the secret hideout.
Youtube video showing all diary entries can be found here: