Abstract: A new model of cosmology, the γδCDM model, is proposed based on an anisotropic background and a specific f(R) theory of gravity. This model is distinguished in two important aspects from the ΛCDM model: firstly, the contribution of different energy densities to the Hubble parameter are weighted with different weights, and then, dependence of energy densities to redshift is modified as well. This unorthodox relation of energy content to the Hubble parameter brings forth a new way of interpreting the cosmological history. This model does not allow the existence of a cosmological constant component; nonetheless, a redshift-dependent dark energy contribution is possible. We tested observational relevance of the new model by best fitting to different data sets. It will be presented that this model could accommodate the idea of cosmological coupling of black holes. It could also bring an explanation to the “impossibly early galaxy problem” by allocating more time for the observed large and massive structures to form from the initial seeds. Thanks to a modified time-redshift relation, the γδCDM model has a more plausible time period at high redshift for large and massive galaxies and massive quasars to form, whereas the age of the Universe today is not modified significantly. We will also show that the Hubble constant is a true constant in this model, it does not evolve with redshift, unlike in the case of the ΛCDM model. The value of the Hubble constant obtained in the γδCDM model is in the 1σ bound of the late Universe observations.