What are the important geometric-topological consequences of 4-dimensional version of Gauss-Bonnet-Chern theorem?

One can find some applications in Arthur Besse's book "Einstein manifolds", $\S6\ D$.

Proposition. If a closed manifold $M^4$ admits an Einstein metric, then $\chi(M)\ge 0$. Moreover $\chi(M)=0$ iff $M$ admits a flat metric.

Proof. Since $Rc(g)=\lambda g$, we have $|Rc|^2=4\lambda^2=r^2/4$. This implies $\chi(M)=1/32\pi^2 \int|Rm|^2dV\ge 0$ with equality iff $Rm=0$. $\square$

Combining with another formula for the signature of a 4-manifold $$ \sigma(M)=\frac{1}{12\pi^2}\int_M(|W_+|^2-|W_-|^2)dV, $$ where $W_{\pm}$ are compontens of the Weyl tensor, one can prove a stronger inequality: for a closed Einstein manifold $M^4$ $$ \chi(M)\ge 3/2|\sigma(M)|. $$

The equality case was studied by Nigel Hitchin, Compact four-dimensional Einstein manifolds, (1974). He proved that if $(M^4, g)$ is an Einstein manifold with $\chi(M)= 3/2|\sigma(M)|$, then $M$ is Ricci flat and either universal cover of $M$ is K3-surface, or $M$ is flat.

These are important results, since topological obstructions for the existence of Einstein metrics are very scarce. In particular in dimensions $\ge 5$ there are no known such obstructions.

A good paper in this direction is "Some implications of the generalized Gauss-Bonnet theorem" written by Bishop and Goldberg which proved the following two theorems

Theorem 1.1. A compact and oriented Riemannian manifold of dimension $4$ whose sectional curvatures are non-negative or nonpositive has non-negative Euler-Poincare characteristic. If the sectional curvatures are always positive or always negative, the Euler-Poincare characteristic is positive.

Theorem 1.2. In order that a $4$-dimensional compact and orientable manifold M carry an Einstein metric, i.e., a Riemannian metric of constant Ricci or mean curvature $R$, it is necessary that its Euler-Poincare characteristic be non-negative.

and a corollary

Corollary 1.2. If $V$ is the volume of $M$, $$\chi(M)\geq\frac{VR^2}{12\pi^2}$$ equality holding if and only if $M$ has constant curvature.