  Given the Luna-Vust invariants of a spherical variety, we determine the
Luna-Vust invariants of the spectrum of its Cox ring. In particular, we deduce
an explicit description of the divisor class group of the Cox ring. It follows
that every spherical variety admits iteration of Cox rings. Moreover, we obtain
a combinatorial proof of the fact that the Cox ring is determined by the
spherical skeleton, which is a known result following from the description of
the Cox ring due to Brion. Finally, we show that a conjectural combinatorial
smoothness criterion can be reduced to the case of a factorial affine spherical
variety with a fixed point.
