  The fate of planets around rapidly evolving stars is not well understood.
Previous studies have suggested that relative to the main sequence population,
planets transiting evolved stars ($P$ $<$ 100 d) tend to have more eccentric
orbits. Here we present the discovery of TOI-4582 b, a 0.94 $\pm$ 0.12
R$_\mathrm{J}$, 0.53 $\pm$ 0.05 M$_\mathrm{J}$ planet orbiting an
intermediate-mass subgiant star every 31.034 days. We find that this planet is
also on a significantly eccentric orbit ($e$ = 0.51 $\pm$ 0.05). We then
compare the population of planets found transiting evolved (log$g$ $<$ 3.8)
stars to the population of planets transiting main sequence stars. We find that
the rate at which median orbital eccentricity grows with period is
significantly higher for evolved star systems than for otherwise similar main
sequence systems, particularly for systems with only one planet detected. In
general, we observe that mean planet eccentricity $<e>$ = $a$ +
$b$log$_{10}$($P$) for the evolved population with a single transiting planet
where $a$ = (-0.18 $\pm$ 0.08) and $b$ = (0.38 $\pm$ 0.06), significantly
distinct from the main sequence planetary system population. This trend is seen
even after controlling for stellar mass and metallicity. These systems do not
appear to represent a steady evolution pathway from eccentric, long-period
planetary orbits to circular, short period orbits, as orbital model comparisons
suggest inspiral timescales are uncorrelated with orbital separation or
eccentricity. Characterization of additional evolved planetary systems will
distinguish effects of stellar evolution from those of stellar mass and
composition.
