The objective of this paper is to study static spherically symmetric noncommutative wormhole solutions in the framework of $f(T)$ gravity. We construct $f(T)$ field equations in covariant and effective energy-momentum tensor forms to make a correspondence with general relativity. It is observed that the violation of energy conditions to support the nonstandard wormhole is due to the effective energy-momentum tensor. We explore the noncommutative wormhole solutions for two cases: (i) assume a viable power-law $f(T)$ model to construct the shape function and (ii) a particular shape function is taken to construct the $f(T)$ model. In the first case, only exotic matter forms a wormhole structure in teleparallel gravity, whereas for $f(T)$ gravity, normal matter threads these structures except for a particular range of $r$. For the constructed $f(T)$ model, there does not exist a physically acceptable wormhole solution similar to the teleparallel case.

In this paper, we study the viability and stability of anisotropic compact stars in the context of $$f({\mathcal {Q}})$$f ( Q ) theory, where $${\mathcal {Q}}$$Q is non-metricity scalar. We use Finch–Skea solutions to investigate the physical properties of compact stars. To determine the values of unknown constants, we match internal spacetime with the exterior region at the boundary surface. Furthermore, we study the various physical quantities, including effective matter variables, energy conditions and equation of state parameters inside the considered compact stars. The equilibrium and stability states of the proposed compact stars are examined through the Tolman–Oppenheimer–Volkoff equation, causality condition, Herrera cracking approach and adiabatic index, respectively. It is found that viable and stable compact stars exist in $$f({\mathcal {Q}})$$f ( Q ) theory as all the necessary conditions are satisfied.

The cosmic expansion phenomenon is being studied through the interaction of newly proposed dark energy models (Tsallis, Rényi and Sharma-Mittal holographic dark energy (HDE) models) with cold dark matter in the framework of loop quantum cosmology. We investigate different cosmic implications such as equation of state parameter, squared sound speed and cosmological plane (${\omega }_d$ - ${\omega }_d^{\prime }$ , ${\omega }_d$ and ${\omega }_d^{\prime }$ represent the equation of state (EoS) parameter and its evolution, respectively). It is found that EoS parameter exhibits quintom like behavior of the universe for all three models of HDE. The squared speed of sound represents the stable behavior of Rényi HDE and Sharma-Mittal HDE at the latter epoch while unstable behavior for Tsallis HDE. Moreover, ${\omega }_d$ - ${\omega }_d^{\prime }$ plane lies in the thawing region for all three HDE models.

Recently, Tsallis, Rényi, and Sharma-Mittal entropies have widely been used to study the gravitational and cosmological setups. We consider a flat FRW universe with linear interaction between dark energy and dark matter. We discuss the dark energy models using Tsallis, Rényi, and Sharma-Mittal entropies in the framework of Chern-Simons modified gravity. We explore various cosmological parameters (equation of state parameter, squared sound of speed ) and cosmological plane (ωd-ωd′, where ωd′ is the evolutionary equation of state parameter). It is observed that the equation of state parameter gives quintessence-like nature of the universe in most of the cases. Also, the squared speed of sound shows stability of Tsallis and Rényi dark energy model but unstable behavior for Sharma-Mittal dark energy model. The ωd-ωd′ plane represents the thawing region for all dark energy models.