It's not often that we turn to the dust jackets of obscure old books for definitions, but for the meaning of "Hot Stove League," we could do a lot worse than this:
Ever since baseball began fans have gathered in winter around the old hot stove, either in a barber shop or country store, to talk about the game they love. The Hot Stove League is an unofficial body, and the only requirement for membership is an avid interest in baseball.
That appeared on the inside flap of baseball historian Lee Allen's 1955 classic, "The Hot Stove League." It was an odd title for Allen's book, which was essentially a compendium of stories he'd run across, all of them connected to the lives of professional players, both on and off the field.