The GALEX Ultraviolet Virgo Cluster Survey (GUViCS). VII.: BCG UV upturn and the FUV-NUV color up to redshift 0.35

CONTEXT:At low redshift, early-type galaxies often exhibit a rising flux with decreasing wavelength in the 1000-2500 \AA{} range, called "UV upturn". The origin of this phenomenon is debated, and its evolution with redshift is poorly constrained. The observed GALEX FUV-NUV color can be used to probe the UV upturn approximately to redshift 0.5. AIMS: We provide constraints on the existence of the UV upturn up to redshift $\sim$ 0.4 in the brightest cluster galaxies (BCG) located behind the Virgo cluster, using data from the GUViCS survey. METHODS:We estimate the GALEX far-UV (FUV) and near-UV (NUV) observed magnitudes for BCGs from the maxBCG catalog in the GUViCS fields. We increase the number of nonlocal galaxies identified as BCGs with GALEX photometry from a few tens of galaxies to 166 (64 when restricting this sample to relatively small error bars). We also estimate a central color within a 20 arcsec aperture. By using the $r$-band luminosity from the maxBCG catalog, we can separate blue FUV-NUV due to recent star formation and candidate upturn cases. We use Lick indices to verify their similarity to redshift 0 upturn cases. RESULTS: We clearly detect a population of blue FUV-NUV BCGs in the redshift range 0.10-0.35, vastly improving the existing constraints at these epochs by increasing the number of galaxies studied, and by exploring a redshift range with no previous data (beyond 0.2), spanning one more Gyr in the past. These galaxies bring new constraints that can help distinguish between assumptions concerning the stellar populations causing the UV upturn phenomenon. The existence of a large number of UV upturns around redshift 0.25 favors the existence of a binary channel among the sources proposed in the literature.