Despite to all significant gravitational phenomena discovered and predicted by general theory of relativity, it is not a complete theory of gravity. One of actual approaches towards more complete theory of gravity is its nonlocal modification.
We consider nonlocal modification of the Einstein theory of gravity in framework of the pseudo-Riemannian geometry. The nonlocal term has the form $\,\mathcal{H}(R)\mathcal{F} (\Box)\mathcal{G}(R),$ where $\mathcal{H}$ and $\mathcal{G}$ are differentiable functions of the scalar curvature $R$, and $\mathcal{F} (\Box) = \sum_{n=0}^{\infty}\, f_n\Box^n$ is an analytic function of the d’Alambert operator $\Box$.
Our motivation to modify gravity, in an analytic nonlocal way, comes mainly from string theory and $p$-adic string theory. After consideration of several models of the above-mentioned type, here we deal with $\mathcal{H}(R)=\mathcal{G}(R)= \displaystyle{\sqrt{R-2\,\Lambda}},$ and where $\mathcal{F} (\Box)$ is an analytic function of the d'Alembert operator $\Box$ and also $\Box^{-1}$. Specially, we investigated several classes of functions as scaling factors, and in some of them we found some new exact cosmological solutions. We are paid our attention to the scaling factor of the form $a(t)=At^{\frac{2}{3}}\,e^{\frac{\Lambda}{14}\,t^2}$, and we test validity of obtained solutions with experimental data and their interpretations.