  In this paper, the metric approach of $f(R)$ theory of gravity is used to
investigate the exact vacuum solutions of spatially homogeneous rotating
spacetimes. For this purpose, R is replaced by f(R) in the standard
Einstein-Hilbert action and the set of modified Einstein field equations reduce
to a single equation. We adopt the assumption of constant Ricci scalar which
maybe zero or non-zero. Moreover, the energy density of the non-trivial
solution has been evaluated by using the generalized Landau-Lifshitz
energy-momentum complex in the perspective of f(R) gravity for some appropriate
f(R) model, which turns out to be a constant quantity.
