A class of nonlocal gravity models, where nonlocal term contains an analytic function of the d'Alembert operator $\square$, is considered. For simplicity, these models are considered without matter sector. Related equations of motion for gravitational field $g_{\mu\nu}(x)$ are presented and analyzed for a constant scalar curvature $R$. The corresponding solutions for the cosmological scale factor $a(t)$ of the FLRW universe are found and discussed.