Question

Is the Landau free energy scale-invariant at the critical point?

Answer 1

no mate landau have purchase
it is not the free energy scale-invariant at the critical point

Answer 2

Power laws are scale-invariant. But for a theory itself to be scale-invariant (as Wikipedia claims) the Landau Free energy functional should have a scale-invariant behaviour at the critical point. But the free energy functional is a polynomial in the order parameter and polynomials are not scale-invariant.