Rapid decay property (RD) of discrete groups is a fundamental tool in the study of reduced group C*-algebras, and allows one to compare the operator norm of convolution operators with much simpler l2-norms. This property was studied for discrete quantum groups by means of `quantum RD` and `twisted quantum RD`, and it was an open question whether non-amenable orthogonal free quantum groups satisfy twisted quantum RD. It has turned out that this question has the negative answer, but a weakened RD is always satisfied. Moreover, this weakened RD allows us to get (almost) optimal time for ultracontractivity of heat semigroups. This talk is based on recent joint work with Michael Brannan and Roland Vergnioux.