I have a Facebook app and I would like to monetize it with Adsense search, which means that people could use a "Google Adsense search box" within Facebook page.
Is it allowed?
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Maybe, there seems to be an number of grey areas.
Google Adsense, believe it or not is used by a huge portion of Application Developers on Facebook for many reasons, even though technically Google Adsense, isn’t designed for Social Sites like Facebook, and thus won’t always approve. Applications on Facebook is a complex juggle to manage to make profitable, using Adverts is just one of them. “Virtual Currency” is a popular choice to go in addition, but that's a topic for another Article.
You run the risk of having the account suspended potentially, since Google Adsense TOS does not permit the use of IFrames to display their ads. so you take my advice at your own risk. Don’t over do it with Ads on your Canvas page, I recommend one, at most 2 per page, and the pages which you put the ads on within the frames, ensure their is a lot of content, and not just keywords there.