In this work, we investigate the thermodynamic properties of a non-rotating hairy Bardeen black hole, highlighting deviations from the predictions of standard general relativity due to the presence of additional parameters. Specifically, we analyze the influence of the electric charge $Q $ , the coupling constant $\beta $ , and the model parameter $\eta $ on the black hole’s mass, temperature, sparsity parameter and entropy. While the overall qualitative behavior of these quantities remains consistent, we find that both $Q $ and $\eta $ tend to decrease the mass and temperature, whereas $\beta $ exerts an opposite effect by increasing them. Furthermore, using the entropic force approach, we derive a novel expression for the black hole entropy, which encapsulates the modifications to the underlying gravitational interaction. We then examine how the parameters $Q $ , $\beta $ , and $\eta $ affect the circular motion of photons. Our results show that the radius of stable circular orbits increases with $Q $ and $\eta $ , while the radius of unstable circular orbits decreases with $Q $ and increases with $\beta $ . Additionally, the critical impact parameter is found to grow with increasing $Q $ , but diminish with increasing $\beta $ . We also study the variations in the Keplerian frequency of photons orbiting the black hole under the influence of these parameters. For small radial distances $r $ , the charge $Q $ reduces the frequency, while for intermediate and large $r $ , it causes an increase. A similar trend is observed for the model parameter $\eta $ for small values of $r$ , whereas the coupling constant $\beta $ produces the opposite effect across these regimes. At the end of the paper, we derive the modified Friedmann equation from the entropy of the studied black hole.
