  In this paper, we prove that a Gorenstein toric Fano variety $(X, -K_{X})$ is
asymptotically Chow semistable then it is Ding polystable with respect to toric
test configurations (Theorem 1.3). This extends the known result obtained by
others (Theorem 1.2) to the case where $X$ admits Gorenstein singularity. We
also show the additivity of the Mabuchi constant for the product toric Fano
varieties in Proposition 1.5 based on the author's recent work (Ono, Sano and
Yotsutani in arXiv:2305.05924). Applying this formula to certain toric Fano
varieties, we construct infinitely many examples that clarify the difference
between relative K-stability and relative Ding stability in a systematic way
(Proposition 1.4). Finally, we verify relative Chow stability for Gorenstein
toric del Pezzo surfaces using the combinatorial criterion developed in
(Yotsutani and Zhou in Tohoku Math. J. 71 (2019), 495-524.) and specifying the
symmetry of the associated polytopes as well.
