Friedmann equations of the fractal apparent horizon

From a fractal perspective, the entropy bound of gravitational systems undergoes changes. Furthermore, in the cosmological setting, the conservation law of a perfect fluid is also altered in such systems, affecting spatial elements like volume, area, and radius. By applying the first law of thermodynamics and deriving the Friedmann equations, we can gain insight into the evolution of such a fractal cosmos. However, observations continue to necessitate the existence of a dark energy source. To address this, in this article, we have created a novel fractal $\Lambda$CDM cosmological model and determined the fractal cosmological observables. We show that the spatial fractal dimension is two, and the age of the Universe is 13.91 Gyr, by fitting the model's parameters to cosmological data.