A~finite simple graph $G$ with a subgraph $H$ is called a super $(b,d)$-$H$-antimagic: if $G$ has an edge covering by subgraphs $H_1, H_2, \dots, H_t$ with each $H_i\cong H$, $i=1, 2, \dots, t$, a total labeling $\alpha$ such that $wt_{\alpha}H$, constitutes an arithmetic progression and $\alpha(V(G))$ consists of the smallest possible integers. In this manuscript, we investigated the existence of super $(b,1)$-star-antimagic labeling of Sun graphs $SG_n^p$.