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Radio telescopes with dual linearly polarized feeds regularly participate in Very Long Baseline Interferometry. One example is the VLBI Global Observing System (VGOS), which is employed for high‐precision geodesy and astrometry. In order to achieve the maximum signal‐to‐noise ratio, the visibilities of all four polarization products are combined to Stokes I before fringe‐fitting. Our aim is to improve cross‐polarization bandpass calibration, which is an essential processing step in this context. Here we investigate the shapes of these station‐specific quantities as a function of frequency and time. We observed the extra‐galactic source 4C 39.25 for 6 hours with a VGOS network. We correlated the data with the DiFX software and analyzed the visibilities with PolConvert to determine the complex cross‐bandpasses with high accuracy. Their frequency‐dependent shape is to first order characterized by a group delay between the two orthogonal polarizations, in the order of several hundred picoseconds. We find that this group delay shows systematic variability in the range of a few picoseconds, but can remain stable within this range for several years, as evident from earlier sessions. On top of the linear phase‐frequency relationship there are systematic deviations of several tens of degrees, which in addition are subject to smooth temporal evolution. The antenna cross‐bandpasses are variable on time scales of ∼1 hr, which defines the frequency of necessary calibrator scans. The source 4C 39.25 is confirmed as an excellent cross‐bandpass calibrator. Dedicated surveys are highly encouraged to search for more calibrators of similar quality. Key Points: The new‐generation geodetic radio telescopes observe two orthogonal linear polarization directionsCalibration of the gain differences between the two polarizers is necessary to maximize the signal‐to‐noise ratio of observationsWe investigate these cross‐polarization gain differences and their temporal evolution for selected antennas [ABSTRACT FROM AUTHOR]