The impact of the stellar mass on the giant planet properties is still to be fully understood. Main-Sequence (MS) stars more massive than the Sun remain relatively unexplored in radial velocity (RV) surveys, due to their characteristics that hinder classical RV measurements. Our aim is to characterize the close (up to 2.5 au) giant planet (GP) and brown dwarf (BD) population around AF MS stars and compare this population to stars with different masses. We used the SOPHIE spectrograph located on the 1.93m telescope at Observatoire de Haute-Provence to observe 125 northern, MS AF dwarfs. We used our dedicated SAFIR software to compute the RV and other spectroscopic observables. We characterized the detected sub-stellar companions and computed the GP and BD occurrence rates combining the present SOPHIE survey and a similar HARPS survey. We present new data on two known planetary systems around the F5-6V dwarfs HD16232 and HD113337. For the latter, we report an additional RV variation that might be induced by a second GP on a wider orbit. We also report the detection of fifteen binaries or massive sub-stellar companions with high-amplitude RV variations or long-term RV trends. Based on 225 targets observed with SOPHIE or HARPS, we constraint the BD frequency within 2-3 au around AF stars to be below 4 percents (1-sigma). For Jupiter-mass GP within 2-3 au (periods below 1000 days), we found the occurrence rate to be 3.7 (+3/-1) percents around AF stars with masses below 1.5 solar masses, and to be below 6 percents around AF stars with masses above 1.5 solar masses. For periods smaller than 10 days, we find the GP occurrence rate to be below 3 or 4.5 percents, respectively. Our results are compatible with the GP frequency reported around FGK dwarfs and are compatible with a possible increase of GP orbital periods with the stellar mass as predicted by formation models.
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