Rat fights were popular in all the big cities that Thoreau visited and lecture and worked in--NYC, Boston and Philly.
More from James Dabney McCabe, writing in "Secrets of the Great City" (1868), on http://terriermandotcom.blogspot.com:
"Rats are plentiful along the East River, and Burns has no difficulty in procuring as many as he desires. These and his dogs furnish the entertainment, in which he delights. The principal room of the house is arranged as an amphitheatre. The seats are rough wooden benches, and in the centre is a ring or pit, enclosed by a circular wooden fence, several feet high. A number of rats are turned into this pit, and a dog of the best feral stock is thrown in amongst them. The little creature at once falls to work to kill the rats, bets being made that she will destroy so many rats in a given time. The time is generally 'made' by the little animal. . . . "
A gratuitous mention of a great rat song, the great rat song, by Casey Neill and the Norway Rats: click on Holy Land here.