  In this paper, we propose and analyze a novel third-order low-regularity
trigonometric integrator for the semilinear Klein-Gordon equation in the
$d$-dimensional space with $d=1,2,3$. The integrator is constructed based on
the full use of Duhamel's formula and the technique of twisted function to the
trigonometric integrals. Rigorous error estimates are presented and the
proposed method is shown to have third-order accuracy in the energy space under
a weak regularity requirement in $H^{2}\times H^{1}$. A numerical experiment
shows that the proposed third-order low-regularity integrator is much more
accurate than the well-known exponential integrators of order three for
approximating the Klein-Gordon equation with nonsmooth solutions.
