  The recent development of brand new observational techniques and theoretical
models have greatly advanced the exoplanet research field. Despite significant
achievements, which have allowed the detection of thousands extrasolar systems,
a comprehensive understanding of planetary formation and evolution mechanisms
is still desired. One relevant limitation is given by the accuracy in the
measurements of planet-host star ages. The star GJ 504 has been found to host a
substellar companion whose nature is strongly debated. There has been a recent
difference of opinion in the literature owing to the uncertainty on the age of
the system: a young age of $\sim$ 160 Myr would imply a giant planet as a
companion, but a recent revision pointing to a solar age ($\sim$ 4 Gyr) instead
suggests a brown dwarf. With the aim of shedding light on this debated topic,
we have carried out a high-resolution spectroscopic study of GJ 504 to derive
stellar parameters, metallicity, and abundances of both light and heavy
elements, providing a full chemical characterisation. The main objective is to
infer clues on the evolutionary stage (hence the age) of this system. We
performed a strictly differential (line-by-line) analysis of GJ 504 with
respect to two reference stars, that is the planet-host dwarf $\iota$ Hor and
the subgiant HIP 84827. The former is crucial in this context because its
stellar parameters (hence the evolutionary stage) is well constrained from
asteroseismic observations. Regardless of the zero point offsets, our
differential approach allows us to put tight constraints on the age of GJ 504
with respect to $\iota$ Hor, thereby minimising the internal uncertainties. We
found that the surface gravity of GJ 504 is 0.2 $\pm$ 0.07 dex lower than that
of the main-sequence star $\iota$ Hor, suggesting a past turn-off
evolution[...]
