The page you link says that they drink more often, not necessarily more.

I imagine rich people use alcohol to spice up other entertainment and poor people more often entertain themselves with alcohol only.

You may be right about the use of alcohol as a sole source of entertainment. But the evidence shows that rich people (this source is the UK) drink more, not just more often: http://www.ias.org.uk/Alcohol-knowledge-centre/Socioeconomic...

These are the top Google results for "alcohol consumption by income level", by the way. There's no need to speculate about it without sources as people in this thread are doing.