  This paper aims to study the low dimensional cohomology of Hom-Lie algebras
and q-deformed W(2,2) algebra. We show that the q-deformed W(2,2) algebra is a
Hom-Lie algebra. Also, we establish a one-to-one correspondence between the
equivalence classes of one dimensional central extensions of a Hom-Lie algebra
and its second cohomology group, leading us to determine the second cohomology
group of the q-deformed W(2,2) algebra. In addition, we generalize some results
of derivations of finitely generated Lie algebras with values in graded modules
to Hom-Lie algebras. As application we compute all $\a^k$-derivations and in
particular the first cohomology group of the q-deformed W(2,2) algebra.
