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.