“Dog days” is the shortened version of “the dog days of summer” and refers to define a period of time or an event that is very hot or stagnant.

In this case I’m sure it refers to stagnant, though I have not read the book myself. So “dog days” would be the most boring and slow days of summer.

An example of a “dog day:” Think of how kids are in the summer when they can’t find a friend to play with and/or think of anything to do. Remember that phrase, “I’m sooo bored! There’s nothing to do!” That could be labeled a dog day.

Hoped that helped!