We present a long-term spectral monitoring of the unique double pulsar binary PSR J0737-3039 corresponding to two "Large Programs" performed by XMM-Newton in 2006 and 2011. Spectral variability of pulsar emission in soft X-rays is not evident over 5 years, despite the significant relativistic spin precession in the considered time span (∼ 25°). We provide, for the first time, evidence of hard X-ray emission from the system in the 5-8 keV energy band. The standard spectral analysis was coupled to the energy dependent spatial analysis to confirm this excess, most likely ascribed to iron line emission. The Fe Kα emission line at 6.4-6.97 keV was previously unheard of in non-accreting binary systems and could testify to the presence of a relic disk that survived the supernova explosions that terminated the lives of the double pulsar's stellar progenitors. The existence of a relic disk in this system reinforces speculation about the presence of similar structures around other peculiar classes of isolated neutron stars.