In this paper, we have discussed a spatially homogeneous and anisotropic Bianchi-type $VI_{0}$ space-time in the presence of holographic dark energy with the Hubble horizon as the infrared cut-off within the frame of the non-linear $f(R, L_{m})$ gravity model $f(R, L_{m})=\beta R+ L_{m}^{\alpha }$ , where $\alpha $ and $\beta $ are the free parameters. Then we have obtained the modified Friedmann equations and solved them by presuming an expansion scalar $\theta (t)$ is proportional to the shear scalar $\sigma (t)$ , which yields the analytical relation between metric potentials as $A=B^{n}$ , where $n$ is constant. Also, we adopted the hyperbolic scale factor $a(t)=\sqrt[\eta ]{\sinh {t}}$ , where $\eta >0$ is constant. Then we estimate the best fit values of model parameters by using a Chi-square test coupled with Markov Chain Monte Carlo (MCMC) simulations, based on Observational Hubble Data (OHD) with 31 points and Pantheon+SHOES data with 1701 points. This approach results in the best fit ranges for model parameters as: $\eta = 1.3_{-0.09}^{+0.08}$ , $H_{0}=65.97_{-2.42}^{+2.48}$ km/s/Mpc for OHD and $\eta = 1.2_{-0.04}^{+0.04}$ , $H_{0}=72.77_{-0.26}^{+0.24}$ km/s/Mpc for the Pantheon+SHOES dataset. Our analysis yields a value for $H_{0}$ that closely aligns with the Planck Collaboration’s 2018 estimate of $H_{0} = 67.4 \pm 0.5$ km/s/Mpc. With these imposed constraints, we derived the deceleration parameter $q$ , indicating a shift from a previously decelerating phase to the current phase of accelerating expansion, with $q_{0} = -0.35$ (OHD) as well as $q_{0} = -0.4$ (Pantheon+SHOES). Then we analyzed the nature of the $Om(z)$ diagnostic parameter, which shows that our universe shows quintessence-type behavior. Additionally, we examined the behavior of energy density, pressure, and the equation of state (EoS) parameter using the holographic dark energy (HDE) model, incorporating Hubble’s horizon as the infrared (IR) cutoff. The equation of state parameter $\omega $ in our model demonstrates that the universe is currently in a quintessence phase, where dark energy behaves as a dynamical component rather than a cosmological constant $\omega =-1$ . Subsequently, the stability of our model was evaluated by analyzing the sound speed squared, denoted as $v_{s}^{2}$ . Finally, we examined the properties of different energy conditions in the context of stability analysis. The DEC and WEC support the model’s credibility due to their positive nature, whereas the violation of SEC suggests the universe is undergoing accelerated expansion. Our analysis reveals that the HDE cosmological model aligns well with recent observational research and effectively accounts for the acceleration observed in the late-time universe.
