  One of the fundamental questions in astronomy is how planetary systems form
and evolve. Measuring the planetary occurrence and architecture as a function
of time directly addresses this question. In the fourth paper of the Planets
Across Space and Time (PAST) series, we investigate the occurrence and
architecture of Kepler planetary systems as a function of kinematic age by
using the LAMOST-Gaia-Kepler sample. To isolate the age effect, other stellar
properties (e.g., metallicity) have been controlled. We find the following
results. (1) The fraction of stars with Kepler-like planets ($F_{\text{Kep}}$)
is about 50% for all stars; no significant trend is found between
$F_{\text{Kep}}$ and age. (2) The average planet multiplicity ($\bar{N}_p$)
exhibits a decreasing trend (~2$\sigma$ significance) with age. It decreases
from $\bar{N}_p$~3 for stars younger than 1 Gyr to $\bar{N}_p$~1.8 for stars
about 8 Gyr. (3) The number of planets per star
($\eta=F_{\text{Kep}}\times\bar{N}_p$) also shows a decreasing trend
(~2-3$\sigma$ significance). It decreases from $\eta$~1.6-1.7 for young stars
to $\eta$~1.0 for old stars. (4) The mutual orbital inclination of the planets
($\sigma_{i,k}$) increases from $1.2^{+1.4}_{-0.5}$ to $3.5^{+8.1}_{-2.3}$ as
stars aging from 0.5 to 8 Gyr with a best fit of
$\log{\sigma_{i,k}}=0.2+0.4\times\log{\frac{\text{Age}}{\text{1Gyr}}}$.
Interestingly, the Solar System also fits such a trend. The nearly independence
of $F_{\text{Kep}}$~50% on age implies that planet formation is robust and
stable across the Galaxy history. The age dependence of $\bar{N}_p$ and
$\sigma_{i,k}$ demonstrates planetary architecture is evolving, and planetary
systems generally become dynamically hotter with fewer planets as they age.
