  We give formulas for various valuative invariants of linearized ample line
bundles on a projective spherical variety. Then we show how to use these
formulas to study $g$-solitons on a Fano spherical variety. As a corollary, we
show that for a Fano horospherical manifold $X$, the corresponding cone
$(K_{X})^{\times}$ always admits a Calabi-Yau cone metric.
