  We investigate the possibility of future singularity due to the accelerating
expansion of the Universe. It has been found that the gravitational theory
comprises Ricci scalar $R$ and Gauss-Bonnet invariant $\mathcal{G}$, known as
$F(R,\mathcal{G})$ gravity, which can be viewed in the quadratic form. Three
models are presented using Hubble parameters to represent a finite and infinite
future. The parameters of the models are analyzed on the basis of their
physical and geometrical properties. This study also explores the properties of
the modified gravitational theory, and neither the future singularity nor the
little or pseudo rip is posed as threats to the fate of the Universe. We
present scalar perturbation approaches to perturbed evolution equations and
demonstrate their stability.

General Relativity and Quantum Cosmology;High Energy Physics - Theory