Aims. The aim of this paper is to investigate the transition of a strongly magnetized neutron star into the accretion regime with very low accretion rate.

Methods. For this purpose, we monitored the Be-transient X-ray pulsar GRO J1008-57 throughout a full orbital cycle. The current observational campaign was performed with the Swift/XRT telescope in the soft X-ray band (0.5-10 keV) between two subsequent Type I outbursts in January and September 2016.

Results. The expected transition to the propeller regime was not observed. However, transitions between different regimes of accretion were detected. In particular, after an outburst, the source entered a stable accretion state characterised by an accretion rate of similar to 10(14)-10(15) g s(-1). We associate this state with accretion from a cold (low-ionised) disc of temperature below similar to 6500 K. We argue that a transition to this accretion regime should be observed in all X-ray pulsars that have a certain combination of the rotation frequency and magnetic field strength. The proposed model of accretion from a cold disc is able to explain several puzzling observational properties of X-ray pulsars.