  We study the nonlinear wave equation (NLW) on the two-dimensional torus
$\mathbb T^2$ with Gaussian random initial data on $H^s(\mathbb T^2) \times
H^{s-1}(\mathbb T^2)$, $s < 0$, distributed according to the base Gaussian free
field $\mu$ associated with the invariant Gibbs measure studied by Thomann and
the first author (2020). In particular, we investigate the approximation
property of the corresponding solution by smooth (random) solutions. Our main
results in this paper are two-fold. (i) We show that the solution map for the
renormalized cubic NLW defined on the Gaussian free field $\mu$ is the unique
extension of the solution map defined for smoothed Gaussian initial data
obtained by mollification, independent of mollification kernels. (ii) We also
show that there is a regularization of the Gaussian initial data so that the
corresponding smooth solutions almost surely have no limit in the natural
topology. This second result in particular states that one can not use
arbitrary smooth approximation for the renormalized cubic NLW dynamics.
  As a preliminary step for proving (ii), we establish a (deterministic) norm
inflation result at general initial data for the (unrenormalized) cubic NLW on
$\mathbb T^d$ and $\mathbb R^d$ in negative Sobolev spaces, extending the norm
inflation result by Christ, Colliander, and Tao (2003).
