We develop the n-dimensional cosmology for $$f(\mathcal {G})$$ $f\left(G\right)$ gravity, where $$\mathcal {G}$$ $G$ is the Gauss–Bonnet topological invariant. Specifically, by the so-called Noether Symmetry Approach, we select $$f(\mathcal {G})\simeq \mathcal {G}^k$$ $f\left(G\right)\simeq G^k$ power-law models where k is a real number. In particular, the case $$k = 1/2$$ $k=1/2$ for $$n=4$$ $n=4$ results equivalent to General Relativity showing that we do not need to impose the action $$R+f(\mathcal {G})$$ $R+f\left(G\right)$ to reproduce the Einstein theory. As a further result, de Sitter solutions are recovered in the case where $$f(\mathcal {G})$$ $f\left(G\right)$ is non-minimally coupled to a scalar field. This means that issues like inflation and dark energy can be addressed in this framework. Finally, we develop the Hamiltonian formalism for the related minisuperspace and discuss the quantum cosmology for this model.