I'll get to books using this title perhaps a bit later.
The NYPL's Lincoln Center Dance Collection had one clipping about a
dance with this "Gay Life" title. It's from the Hearst Sunday Magazine (NY
American, LA Examiner, and other newspapers), copyright 1911. A November
1911 date is in an article on the other side of the clipping.
"Gay Life" is "An erring woman's dance, portraying her career from the
dawn of girlish, innocent beauty, through the old, familiar stages to
inevitable despair and death. This is the dance by Louis La Gai, which is
said to have 'made Paris Good.' (...) It is said that the dance by Louise La
Gai (Louis or Louise?--ed.), at the Theatre Marigny, sent gay Parisian
audiences away with tears in their eyes, because of the terrible vividness
with which it depicted the temptations which may assail any woman, and
because it showed that for the woman to yield meant no end but that of a
despairing death. The effect was, perhaps, all the keener because the dance
was programmed 'The Gay Life.'"
For whatever this is worth, it's a "Gay Life" dance and a choreographer
named "Gai" in "Gay Paree."
Incidentally, OED has "Gay Paree" for 1930.