Special Lagrangian cones in complex Euclidean spaces are
obtained as cones over compact minimal Legendrian submanifolds
in the odd dimenisonal standard hypersphere. The notion of
the stability, the Legendrian stability and the rigidity of
special Lagrangian cones were recently introduced and investigated
by D. Joyce, M. Haskins etc. In this paper we determine explicitly
the stability-index, the Legendrian-index, and the rigidity
of special Lagrangian cones over compact irreducible symmeric
spaces of type $A$ obtained as minimal Legendrian orbits and
over a minimal Legendrian $\mathit{SU}(2)$-orbit. We obtain
the examples of stable and rigid special Lagrangian cones
in higher dimensions. Moreover we discuss a relationship of
these properties with the Hamiltonian stability of minimal
Lagrangian submanifolds in complex projective spaces.