  In this work we establish weak convergence rates for temporal discretisations
of stochastic wave equations with multiplicative noise, in particular, for the
hyperbolic Anderson model. For this class of stochastic partial differential
equations the weak convergence rates we obtain are indeed twice the known
strong rates. To the best of our knowledge, our findings are the first in the
scientific literature which provide essentially sharp weak convergence rates
for temporal discretisations of stochastic wave equations with multiplicative
noise. Key ideas of our proof are a sophisticated splitting of the error and
applications of the recently introduced mild It\^{o} formula. We complement our
analytical findings by means of numerical simulations in Python for the decay
of the weak approximation error for SPDEs for four different test functions.
