In this paper, we consider the following one-dimensional Schnakenberg model with periodic heterogeneity: \begin{document}$ \begin{equation*} \begin{cases} u_t-\varepsilon ^2 u_{xx} = d\varepsilon -u+g(x)u^2 v , & x \in (-1,1) ,\; t>0, \varepsilon v_t-Dv_{xx} = \frac{1}{2}-\frac{c}{\varepsilon}g(x)u^2 v , & x \in (-1,1) ,\; t>0, u_x (\pm 1) = v_x (\pm 1) = 0 .\end{cases} \end{equation*} $\end{document} where \begin{document}$ d,c,D>0 $\end{document} are given constants, \begin{document}$ \varepsilon >0 $\end{document} is sufficiently small, and \begin{document}$ g(x) $\end{document} is a given positive function. Let \begin{document}$ N \ge 1 $\end{document} be an arbitrary natural number. We assume that \begin{document}$ g(x) $\end{document} is a periodic and symmetric function, namely \begin{document}$ g(x) = g(-x) $\end{document} and \begin{document}$ g(x) = g(x+2N^{-1}) $\end{document} . We study the stability of \begin{document}$ N $\end{document} -peak stationary symmetric solutions. In particular, we are interested in the effect of the periodic heterogeneity \begin{document}$ g(x) $\end{document} above on their stability. For the standard Schnakenberg model, namely the case of \begin{document}$ g(x) = 1 $\end{document} , with \begin{document}$ d = 0 $\end{document} , the stability of \begin{document}$ N $\end{document} -peak solutions was established by Iron, Wei, and Winter in 2004. In this paper, we rigorously give a linear stability analysis and reveal the effect of the periodic heterogeneity on the stability of \begin{document}$ N $\end{document} -peak solution. In particular, we investigate how \begin{document}$ N $\end{document} -peak solutions is stabilized or destabilized by the effect of periodic heterogeneity compared with the case \begin{document}$ g(x) = 1 $\end{document} .