Centre of rigidity is a measure of 'torsional susceptibility' of buildings, and is computed by calculating stiffness of vertical members, at a particular floor level, to lateral loads. Hence, in elastic range, it is independent of loading.
But loading can change the state of member: can push a member into inelastic region. In in-elastic region, center of rigidity will change continuously because stiffness of some members will diminish, depending on the state of members and their mechanical properties, which will depend on loading condition.
In design offices, and as far as building codes is concerned, you, primarily, deal with elastic response, and hence it is independent of loading.