Let \(\mathcal{M}_1(M^n)\) be the space of smooth Riemannian metrics of volume one on closed Riemannian manifold \(M^n\), \(n\geq 3\). It is well known that Einstein metrics are critical for the Einstein-Hilbert scalar curvature functional. Then, it is natural to study critical metrics which arise as solutions of the Euler-Lagrange equations for more general curvature functionals or even high order curvature functionals. In this paper, the authors give some new characterizations on critical metrics for the functional \(\mathcal{F}_t= \int_M |\mathrm{Ric}|^2dv+t\int_M R^2 dv\) on \(\mathcal{M}_1(M^n)\), where \(t\in \mathbb{R}\), and Ric and \(R\) denote the Ricci curvature and the scalar curvature, respectively. There exist critical metrics of \(\mathcal{F}_t\) which are not necessarily Einstein. It is natural to ask under what conditions a critical metric for \(\mathcal{F}_t\) must be Einstein. In this article the authors also obtain the conditions on the curvature under which the critical metrics are Einstein.