  We discrete the ergodic semilinear stochastic partial differential equations
in space dimension $d \leq 3$ with additive noise, spatially by a spectral
Galerkin method and temporally by an exponential Euler scheme. It is shown that
both the spatial semi-discretization and the spatio-temporal full
discretization are ergodic. Further, convergence orders of the numerical
invariant measures, depending on the regularity of noise, are recovered based
on an easy time-independent weak error analysis without relying on Malliavin
calculus. To be precise, the convergence order is $1-\epsilon$ in space and
$\frac{1}{2}-\epsilon$ in time for the space-time white noise case and
$2-\epsilon$ in space and $1-\epsilon$ in time for the trace class noise case
in space dimension $d = 1$, with arbitrarily small $\epsilon>0$. Numerical
results are finally reported to confirm these theoretical findings.
