On a Riemannian manifold $M$ with riemann curvature tensor $R\_{\mu\nu\rho\sigma}$ written as (endomorphism valued) curvature two-tensor of the Levi-Civita connection $R=R\_{\mu\nu}dx^\mu\wedge dx^\nu$, consider the following term. 
 $$\int\_M tr(R\wedge \*R)=2\int\_M tr(R\_{\mu\nu}R^{\mu\nu})dvol\_g=2\int\_M tr(R\_{\mu\nu}{}\_{\rho\sigma}R^{\mu\nu\sigma\rho})dvol\_g$$
This is Yang-Mills action evaluated on Levi-Civita. My questions are 


What is its meaning? Is it a topological term?


Does it have any physical applications, especially in GR?



What is the meaning of Yang-Mills action evaluated on Levi-Civita connection?
You can find information about functionals that are quadratic in the curvature in Besse's book *Einstein Manifolds*. In particular, see Chapter 4, Section H and the references cited therein. Your questions about topological interpretations and possible connections with GR, etc., are addressed in some of those references.


