LRS Bianchi type-I with Hubble’s horizon as IR cut-off in f(R) gravity

we explored the holographic dark energy model using Hubble’s Horizon as the infrared (IR) cut-off in Locally Rotationally Symmetric (LRS) Bianchi type-I, considering the $f\left(R\right)$ gravity framework in our analysis. In order to solve the field equations, we assume a relation $F\propto a^m$ and volumetric power law expansion (where $m$ is constant). we have expressed various crucial cosmological parameters in terms of the redshift $z$ and depicted them graphically to enhance our understanding of the expansion and evolution of the universe like holographic dark energy density (${\rho }_{\Lambda }$), holographic dark energy pressure ($p_{\Lambda }$), equation of state parameter ($\omega$) , total energy density parameter ($\Omega$) etc. Also, we analyzed the stability of the universe in our model through the squared speed of sound test and its validity by energy conditions. Ultimately, our model indicates that the universe is currently in an expanding phase, exhibiting an accelerating phase, closely approaching a flat geometry, and its behavior resembles that of a quintessence dark energy model.