We enumerate a framework for optimal transportation on Lie Groups using costs that depend on geodesic distances derived from spatially varying data-driven metrics. We build on the entropic regularized formulation which can be efficiently solved using Sinkhorn iterations. We estimate local distances on the Lie group using logarithmic distance approximations and formulate their extension to a more general setting of data-driven metric tensors. Our formulation leads to a data-driven approximation of the Gibbs kernel which is essential to the Sinkhorn framework. We demonstrate our method with two experiments: Tractography with Diffusion-Weighted MRI and crossing-preserving interpolations of measures in SE(2).