The Double Stout gets some of it inspiration from the Russian Imperial Stout. American brewers, inclined as they are to experiment with the limits of beer styles, have taken the American Stout to bigger levels of malt and hop character. Thus, they've “doubled,” or “imperialized,” the American Stout into a stronger brew. Expect a very robust, full-bodied Stout with lots of complexity in aroma and flavor, with many examples surpassing Russian Imperial Stouts in strength and intensity. Look for a dark, virtually black color displaying the beer’s roasted malt soul.
The rich malts combine with generally high hop aromas and flavors along with alcohol notes and esters of dark fruits while some are infused with coffee or chocolate. All elements are typically balanced and exhibit a layered complexity, with many examples aged in whiskey or bourbon barrels for additional levels of aromas and flavor character. They also may be cellar-aged during which the beer’s character will evolve. Alcohol ranges vary, but tend to be quite big, and bigger than traditional Russian Imperial Stouts. These are beers for sipping and savoring.
As this beer is such a close cousin to the Russian Imperial Stout you can read more about this style here.
Source: beeradvocate - http://www.beeradvocate.com/beer/style/157/, totalwine - http://www.totalwine.com/eng/guide-to-beer/american-double-imperial-stout.cfm