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We analyze new XMM-Newton and archival Chandra observations of themiddle-aged $\gamma$-ray radio-quiet pulsar J1957+5033. We detect, for thefirst time, X-ray pulsations with the pulsar spin period of the point-likesource coinciding by position with the pulsar. This confirms the pulsar natureof the source. In the 0.15--0.5 keV band, there is a single pulse per periodand the pulsed fraction is $\approx18\pm6$ per cent. In this band, the pulsarspectrum is dominated by a thermal emission component that likely comes fromthe entire surface of the neutron star, while at higher energies ($\gtrsim0.7$keV) it is described by a power law with the photon index $\Gamma \approx 1.6$.We construct new hydrogen atmosphere models for neutron stars with dipolemagnetic fields and non-uniform surface temperature distributions withrelatively low effective temperatures. We use them in the spectral analysis andderive the pulsar average effective temperature of $\approx(2-3)\times10^5$ K.This makes J1957+5033 the coldest among all known thermally emitting neutronstars with ages below 1 Myr. Using the interstellar extinction--distancerelation, we constrain the distance to the pulsar in the range of 0.1--1 kpc.We compare the obtained X-ray thermal luminosity with those for other neutronstars and various neutron star cooling models and set some constraints onlatter. We observe a faint trail-like feature, elongated $\sim 8$ arcmin fromJ1957+5033. Its spectrum can be described by a power law with a photon index$\Gamma=1.9\pm0.5$ suggesting that it is likely a pulsar wind nebula powered byJ1957+5033.

Middle aged $\gamma$-ray pulsar J1957+5033 in X-rays: pulsations, thermal emission and nebula