Growth of Cosmic Structures in generalized mass-to-horizon relation Entropic Cosmology

We investigate the growth of cosmic structures in the thermodynamically consistent generalised mass-to-horizon entropic cosmology (MHEC). For the Bekenstein case the entropic energy density augments the Friedmann equations without modifying the Hawking temperature and automatically satisfies the Clausius relation, thereby avoiding the inconsistencies that afflicted earlier entropy-force models. We then derive the linear perturbation equations, emphasising the distinction between a fully perturbed interaction term and the common approximation in which the perturbation is neglected. Numerical solutions show that fully perturbed follows the LCDM matter-growth history within the current growth uncertainties. Our results demonstrate that MHEC matches both background and growth probes as well as LCDM without extra free parameters, providing a viable entropic explanation for recent accelerated expansion of the Universe.