Context. Giant planets grow and acquire their gas envelope during the disk phase. At the time of the discovery of giant planets in their host disk, it is important to understand the interplay between the host disk and the envelope and circum-planetary disk properties of the planet. Aims: Our aim is to investigate the dynamical and physical structure of the gas in the vicinity of a Jupiter-mass planet and study how protoplanetary disk properties, such as disk mass and viscosity, determine the planetary system as well as the accretion rate inside the planet’s Hill sphere. Methods: We ran global 3D simulations with the grid-based code fargOCA, using a fully radiative equation of state and a dust-to-gas ratio of 0.01. We built a consistent disk structure starting from vertical thermal equilibrium obtained by including stellar irradiation. We then let a gap open with a sequence of phases, whereby we deepened the potential and increased the resolution in the planet’s neighbourhood. We explored three models. The nominal one features a disk with surface density, ∑, corresponding to the minimum mass solar nebula at the planet’s location (5.2 au), characterised by an α viscosity value of 4 10$^{−3}$ at the planet’s location. The second model has a surface density that is ten times smaller than the nominal one and the same viscosity. In the third model, we also reduced the viscosity value by a factor of 10. Results: During gap formation, giant planets accrete gas inside the Hill sphere from the local reservoir. Gas is heated by compression and cools according to opacity, density, and temperature values. This process determine the thermal energy budget inside the Hill sphere. In the analysis of our disks, we find that the gas flowing into the Hill sphere is approximately scaled as the product ∑_ν, as expected from viscous transport. The accretion rate of the planetary system (envelope plus circum-planetary disk) is instead scaled as √Σ_ν, with its efficiency depending on the thermal energy budget inside the Hill sphere. Conclusions: Previous studies have shown that pressure-supported or rotationally supported structures are formed around giant planets, depending on the equation of state (EoS) or on the opacity; namely, on the dust content within the Hill sphere. In the case of a fully radiative EoS and a constant dust to gas ratio of 0.01, we find that low-mass and low-viscosity circum-stellar disks favour the formation of a rotationally supported circum-planetary disk. Gas accretion leading to the doubling time of the planetary system of > 105 years has only been found in the case of a low-viscosity disk.