We consider f(Q) extended symmetric teleparallel cosmologies, where Q is the non-metricity scalar, and constrain its functional form through the order reduction method. By using this technique, we are able to reduce and integrate the field equations and thus to select the corresponding models giving rise to bouncing cosmology. The selected Lagrangian is then used to develop the Hamiltonian formalism and to obtain the Wave Function of the Universe which suggests that classical observable universes can be recovered according to the Hartle Criterion.

f(Q) symmetric-teleparallel gravity is considered in view of quantum cosmology. Specifically, we derive cosmological equations for f(Q) models and then investigate the related energy conditions. In the Minisuperspace formalism, the point-like f(Q) Hamiltonian is taken into account. In this framework, we obtain and solve the Wheeler–De Witt equation, thus finding the wave function of the universe in different cases. We show that the Hartle criterion can be applied and classical observable universes occur.

We study a theory of gravity of the form $f\left(\mathcal{G}\right)$ where $\mathcal{G}$ is the Gauss–Bonnet topological invariant without considering the standard Einstein–Hilbert term as common in the literature, in arbitrary $(d+1)$ dimensions. The approach is motivated by the fact that, in particular conditions, the Ricci curvature scalar can be easily recovered and then a pure $f\left(\mathcal{G}\right)$ gravity can be considered a further generalization of General Relativity like $f\left(R\right)$ gravity. Searching for Noether symmetries, we specify the functional forms invariant under point transformations in a static and spherically symmetric spacetime and, with the help of these symmetries, we find exact solutions showing that Gauss–Bonnet gravity is significant without assuming the Ricci scalar in the action.