This question made me think and revisit some of the apps using this pattern to look specifically for any Usability issues. Some more include - Facebook Messenger, Tumblr, Swarm by Foursquare.
When I had first seen these apps they seemed odd to me as an Android UX designer who is aware of Android guidelines. But I don't think they pose any major usability concerns (though a heuristic that it could violate is consistency).
While taking this design approach (of using Action bar for navigation) an important point to consider would be to define place for contextual actions.
One way is to use a Floating Action Button on that page (As in Tumblr). But then there can be only one such button on a screen. Another approach is as in Facebook Messenger that uses a bottom bar for contextual buttons.
Also it might be useful to use the Action bar to describe each screen as in Facebook's primary app.
I think going forward as more apps are released with emphasis on content first approach instead of deep navigations, the Action Bar would become less specific.
In fact Google Fit does not have an Action Bar and most of the functionality is available through Action Overflow.