Recent economics theories that involve differential topology? The original development of general equilibrium theories involved differential topology. 

I wonder if there are any recently developed theories, in any field of economic theories, that utilize differential topology (DT).

I've seen some recent papers but none of them used DT. Some real analysis, topological topology, and convex analysis seem enough.
The main reason differential topology had some success in economics is that supplies powerful methods to show that something holds generically, mainly Sard's theorem and the transversality theorem. Some of these methods have been generalized to contexts without differentiability, see for example the paper "A Prevalent Transversality Theorem for Lipschitz Functions" by Shannon. Differential topology is still used in general equilibrium theory, you might take a look at recent work of Yves Balasko. 

In positive political theory, there are results to the effect that generically no majority winner exists in multidimensional spatial voting models.  A good introduction to this topic is the book "Positive Political Theory I" by Austen-Smith and Banks, the definitive treatment is the paper "The generic existence of a core for q-rules" by Saari. 

Inspired by the work on regular equilibria in general equilibrium theory, regular Nash equilibria were also studied in game theory, starting with Harsanyi. An elegant approach can be found in the paper "The theory of normal form games from the differentiable viewpoint" by Ritzberger (preprint here). Many genericity results in general equilibrium have corresponding versions in game theory, for example, "most" normal form games have a finite and odd number of Nash equilibria.

Another area where differential topology is used is in the study of stability with respect to some dynamics. Indeed, one can define regular equilibria via the replicator dynamic from evolutionary game theory (that is how Ritzberger does it). For a recent example of using some differential topology (index theory, treat also in more generality), see the working paper "The Index +1 Principle" by McLennan.