  We propose a new model of cosmology based on an anisotropic background and a
specific $f(R)$ theory of gravity. It is shown that field equations of $f(R)$
gravity in a Bianchi type I background give rise to a modified Friedmann
equation. This model contains two important parameters: $\gamma$ and $\delta$.
We, thus, simply call our model $\gamma\delta$CDM. It is distinguished in two
important aspects from the $\Lambda$CDM 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 Hubble parameter brings
forth a new way of interpreting the cosmological history. This solution does
not allow the existence of a cosmological constant component, however, a dark
energy contribution with dependence on redshift is possible. We tested
observational relevance of the new solution by best fitting to different data
sets. We found that our model could accommodate the idea of cosmological
coupling of black holes.
