How to use Gauss-Bonnet theorem on a cube

Question

The Gauss-Bonnet theorem for compact regular surfaces is often only enunciated for totally regular surfaces, with no singular points. But what if I wanted to state it on a surface with a finite number of singular points but otherwise regular, like a cube? How is the Gauss-Bonnet modified? I know somehow the curvature has to be concentrated on the vertices, but I'd like some help in making that notion precise.
Thank you.