Super-Earths are by far the most dominant type of exoplanet, yet their formation is still not well understood. In particular, planet formation models predict that many of them should have accreted enough gas to become gas giants... Here we examine the role of the protoplanetary disk in the cooling and contraction of the protoplanetary envelope. In particular, we investigate the effects of 1) the thermal state of the disk as set by the relative size of heating by accretion or irradiation, and whether its energy is transported by radiation or convection, and 2) advection of entropy into the outer envelope by disk flows that penetrate the Hill sphere, as found in 3D global simulations. We find that, at 5 and 1 AU, this flow at the level reported in the non-isothermal simulations where it penetrates only to ~ 0.3 times the Hill radius has little effect on the cooling rate since most of the envelope mass is concentrated close to the core, and far from the flow. On the other hand, at 0.1 AU, the envelope quickly becomes fully-radiative, nearly isothermal, and thus cannot cool down, stalling gas accretion. This effect is significantly more pronounced in convective disks, leading to envelope mass orders of magnitude lower. Entropy advection at 0.1 AU in either radiative or convective disks could therefore explain why super-Earths failed to undergo runaway accretion. These results highlight the importance of the conditions and energy transport in the protoplanetary disk for the accretion of planetary envelopes. (read more)