A bouncing scenario of a flat homogeneous and isotropic universe is explored by using the reconstruction technique for the power-law parametrization of the Hubble parameter in a modified gravity theory with higher-order curvature and trace of the energy-momentum tensor terms. It is demonstrated that bouncing criteria are satisfied so that the cosmological initial singularity can be avoided. In addition, it is shown that the equation of state parameter crosses the line of the phantom divide. In the present scenario, the universe is filled with perfect fluid around the bouncing point, in which the universe becomes highly unstable and a big bounce can be realized. Furthermore, it is found that extremal acceleration occurs at the bouncing point.

We propose a novel dark-energy equation-of-state parametrization, with a single parameter $\eta$ that quantifies the deviation from $\Lambda$CDM cosmology. We first confront the scenario with various datasets, from the Hubble function (OHD), Pantheon, baryon acoustic oscillations (BAO), and their joint observations, and we show that $\eta$ has a preference for a non-zero value, namely, a deviation from $\Lambda$CDM cosmology is favored, although the zero value is marginally inside the 1$\sigma$ confidence level. However, we find that the present Hubble function value acquires a higher value, namely, $H_0=66.{624}_{-0.013}^{+0.011}$ Km s⁻¹ Mpc⁻¹, which implies that the $H_0$ tension can be partially alleviated. Additionally, we perform a cosmographic analysis, showing that the universe transits from deceleration to acceleration in the recent cosmological past; nevertheless, in the future, it will not result in a de Sitter phase since it exhibits a second transition from acceleration to deceleration. Finally, we perform the statefinder analysis. The scenario behaves similarly to the $\Lambda$CDM paradigm at high redshifts, while the deviation becomes significant at late and recent times and especially in the future.

We perform observational confrontation and cosmographic analysis of $f(T,T_G)$ gravity and cosmology. This higher-order torsional gravity is based on both the torsion scalar, as well as on the teleparallel equivalent of the Gauss–Bonnet combination, and gives rise to an effective dark-energy sector which depends on the extra torsion contributions. We employ observational data from the Hubble function and supernova Type Ia Pantheon datasets, applying a Markov chain Monte Carlo sampling technique, and we provide the iso-likelihood contours, as well as the best-fit values for the parameters of the power-law model, an ansatz which is expected to be a good approximation of most realistic deviations from general relativity. Additionally, we reconstruct the effective dark-energy equation-of-state parameter, which exhibits a quintessence-like behavior, while in the future the Universe enters into the phantom regime, before it tends asymptotically to the cosmological constant value. Furthermore, we perform a detailed cosmographic analysis, examining the deceleration, jerk, snap, and lerk parameters, showing that the transition to acceleration occurs in the redshift range $0.52\le z_{tr}\le 0.89$, as well as the preference of the scenario for quintessence-like behavior. Finally, we apply the Om diagnostic analysis to cross-verify the behavior of the obtained model.