TESS Giants Transiting Giants III: An eccentric warm Jupiter supports a period-eccentricity relation for giant planets transiting evolved stars

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.