The Hodge * operator action on cohomology is generally speaking metric-dependent, hence * is not well-defined without fixing the metric. There are some caveats. On complex curves, for example, the Hodge * operator is complex rotation, which depends only on complex structure. This gives an example of a manifold for which the action of * is metric-dependent: indeed, the action of complex rotation on $H^1$ determines the biholomorphism class of the complex curve (Torelli).
On compact 2n-dimensional manifolds, the *-operator in the middle cohomology is determined by the conformal structure.