Summary: Let \(\{\phi_j(z): j\ge 1\}\) be an orthonormal basis of Hecke-Maass cusp forms with Laplace eigenvalue \(1/4+t^2_j\). For each \(\phi_j(z)\), we have the automorphic \(L\)-function \(L(s, \mathrm{sym}^2\phi_j)\) which is called the symmetric square \(L\)-function associated to \(\phi_j\). In this paper, the average estimate of \(L(s, \mathrm{sym}^2\phi_j)\) is considered, i.e., for sufficiently large \(T\), the asymptotic formula \[\sum\limits_j\mathrm{e}^{-\frac{t^2_j}{T^2}}\alpha_jL(1/2,\mathrm{sym}^2\phi_j)=T^2P(\log T)+O(T^{\frac 32+\varepsilon}),\] is established, where \(\alpha_j\) is a certain fixed weight function and \(P(x)\) is a polynomial of degree 1.