  Let $G$ be a connected, complex reductive Lie group and $X$ a $\mathbb
Q$-Fano $G$-spherical variety. In this paper we compute the weighed
non-Archimedean functionals of a $G$-equivariant normal test configurations of
$X$ via combinatory data. Also we define a modified Futaki invariant with
respect to the weight $g$, and give an expression in terms of intersection
numbers. Finally we show the equivalence of different notations of stability
and gives a stability criterion on $\mathbb Q$-Fano spherical varieties, which
is also a criterion of existence of K\"ahler-Ricci $g$-solitons.
