Optimal transport provides an intuitive and robust way to compare probability measures with applications in many areas of mathematics. This holds in particular for the Wasserstein-2 distance with its formal Riemannian structure. While entropic regularization of optimal transport has several favourable effects, such as improved statistical sample complexity, it destroys this metric structure. The de-biased Sinkhorn divergence is a partial remedy, as it is positive, definite, and its sublevel sets induce the weak* topology. However, it does not satisfy the triangle inequality. We resolve this issue by considering the Hessian of the Sinkhorn divergence as a Riemannian tensor and study the induced distance. In this talk we outline the key steps of this construction, the corresponding induced notion of tangent space, some early results on the distance, and open directions for future work.