Exploring phase space trajectories in $\mathrm{\Lambda }$CDM cosmology with $f(G)$ gravity modifications

In this work, the cosmic solutions, particularly the well-known $\mathrm{\Lambda }$CDM model, are investigated in the framework of the Gauss–Bonnet (GB) gravity, where the gravitational action incorporates the GB invariant function. We utilize a specialized formulation of the deceleration parameter in terms of the Hubble parameter $H$, given by $q=-1-\frac{Ḣ}{H^2}$, to solve the field equations. To identify the appropriate model parameters, we align them to the most recent observational datasets, which include 31 data points from the Cosmic Chronometers, Pantheon+, and BAO datasets. The physical characteristics of the cosmographic parameters, such as pressure and energy density, that correlate to the limited values of the model parameters, are examined. The evolution of the deceleration parameter suggests a transition from a decelerated to an accelerated phase of the universe. Additionally, we examine the stability of the assumed model and provide an explanation for late-time acceleration using the energy conditions. The behavior of the equation of state parameter has been analyzed through dynamical variables by constraining various parameters in light of the recent observational data. This study has resulted in a quintessence-like evolution.