What does "that Frank Sinatra at the bar act" mean in this sentence?

When I read a comic book called "Thief of Thieves" by Robert Kirkman, I cannot understand the scene below:

The man is drinking alone at the bar counter and seems deep in thought. And the woman, who was asked by him to meet at the bar, comes and says "Okay, I don't really care how deep you are in that Frank Sinatra at the bar act...you're still buying." They are kind of friends (although their relationship is much more complicated, I just say they are friends to simplify the context).

What does she mean by "that Frank Sinatra at the bar act"? I googled Frank Sinatra and bar but couldn't find other than this medley and Sinatra Piano Bar. Could someone help me with this?

Below is the typical, melancholic image that many Frank Sinatra fans carry of him: sitting at a bar in a trilby hat, a lighted cigarette in one hand, and the perennial glass of hard liquor in the other. Despite the forlorn gaze, Frank Sinatra was one of the best-paid performers in his day, and could easily afford to pay for a couple of drinks.

The line, the Frank Sinatra at the bar act is an original expression, created by the author to suit his specific needs and purposes. The writer could have used hyphens, as suggested by @Lambe in the comments, or used single quotes for clarity. Why he didn't is anyone's guess.

I don't really care how deep you are in that ‘Frank Sinatra at the bar’ act... you're still buying.”

One of the numerous meanings of the noun act cited by Longman Dictionary is the following

3. (Noun) insincere behaviour in which you pretend to have a particular kind of feeling or to be a particular kind of person, e.g. Mike played the loving husband in front of the children but it was all an act.

Collins has

7. (Noun) a show of feeling or behavior that is not sincere and is put on just for effect
- There were moments when I wondered: did she do this on purpose, was it all just a game, an act?