In this paper, we investigate a class of stochastic functional differential equations driven by the time-changed Lévy process. Using the Lyapunov technique, we obtain some sufficient conditions to ensure that the solutions of the considered equations are $h$-stable in $p$-th moment sense. Subsequently, using time-changed Itô formula and a proof by reduction ad absurdum, we capture some new criteria for the $h$-stability in mean square of the considered equations. In the end, we analyze some illustrative examples to show the interest and usefulness of the major results.