On the one hand there are some notions that seem related that are studied (see at the end), but on the other hand the precise defintion you give does not have some, at least from a cetain point of view, desirable features.

First, on this second aspect an example (there are various other 'good' properties of the additive energy but I focus on this one only, here). 

An arithmetic progression is considered as a very "additively closed" set, in some sense it is "as closed as possible"; for example in the sense that (finite) arithmetic progressions are precisley the sets, in the integers (or more generally torsion free abelian groups), $P$ such that $|P+P|=2 |P| -1$ and they are optimal at this as $|A+A| \ge 2 |A| -1$ for any finite set.

Now, your $F(A)$ is indeed small for *some* arithmetic progressions (indeed, I think, as small as possible when fixing the cardinality of the set) but *huge* (as large as possible) for others, (for certain applications) this is not desirable; as an example compare the results for  initial segments of the even and odd natural numbers, respectively. While the additive energy would always be the same for arithemtic progressions, of a given lengths. So I do not think there is much direct relation here.
 
To put this differently it is desirable that such notions are rather invariant under *affine* transformations; in particular, $F(A) = F(b+A)$ would be something that is nice to have. The additive energy has this property, your notion does not.

But, having said that there are investigations in Additive Combinatorics that do go in the direction you seem to have in mind. Specifically, there is the Isoperimetric Method, due to the late Hamidoune, that very roughly speaking is based on considering how much a 'shift' of a set (by certain elements) will differ from the original set, or how much sets 'expand' in certain Cayley graphs.
      
See for example one of his papers ["Topology of Cayley Graphs Applied to Inverse Additive Problems"][1] (but there are many more due to him and others based on this approach; look for example for his papers on arXiv). Note in particular the definition of the boundary of a set there; while this does not match exactly what you envision I thought it is somewhat close in spirit and you thus might find it interesting, and there might be things closer still but unfortunately I do not have a very good overview. 

  


  [1]: http://arxiv.org/abs/1011.1797