  We study the global-in-time dynamics for a stochastic semilinear wave
equation with cubic defocusing nonlinearity and additive noise, posed on the
$2$-dimensional torus. The noise is taken to be slightly more regular than
space-time white noise. In this setting, we show existence and uniqueness of an
invariant measure for the Markov semigroup generated by the flow over an
appropriately chosen Banach space. This extends a result of the second author
to a situation where the invariant measure is not explicitly known.
