Is this cohomology isomorphic to De Rham Cohomology?

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Let $(M,g)$ be a Riemannian manifold. Put $d^{*}= *d*$ where $*$ is the Hodge $*$ operator. So $d^{*}\circ d^{*}=0$. Then it introduce a (c0)homology.

What is a relation between this (co)homology and De Rham cohomology? Are They isomorphic?