  We study algebras encoding stable range branching rules for the pairs of
complex classical groups of the same type in the context of toric degenerations
of spherical varieties. By lifting affine semigroup algebras constructed from
combinatorial data of branching multiplicities, we obtain algebras having
highest weight vectors in multiplicity spaces as their standard monomial type
bases. In particular, we identify a family of distributive lattices and their
associated Hibi algebras which can uniformly describe the stable range
branching algebras for all the pairs we consider.
