Green roofs help reduce heat significantly. For example in this study
Mean daily temperature reductions achieved were 18.0 °C at the soil surface and 27.5 °C below the module
and this EPA study mentions 2 studies:
Chicago compared summertime surface temperatures on a green roof with a
neighboring building. On an August day in the early afternoon, with temperatures in the 90s, the green roof surface temperature ranged from 91 to 119°F (33 to 48°C), while the dark, conventional roof of the adjacent building was 169°F (76°C). The near-surface air temperature above the green roof was about 7°F (4°C) cooler than that over the conventional roof.
A similar study in Florida found that the average maximum surface temperature of a green roof was 86°F (30°C) while the adjacent light-colored roof was 134°F (57°C)
Infrared Image of Chicago City Hall Green Roof and conventional roof, source http://ccap.org/what-does-climate-resilience-look-like
How much temperature reduction you will get is somewhat difficult to predict because it depends on the thickness of the soil layer, how well the grass is growing and most importantly how much moisture is retained (more moisture means more evaporation and larger cooling effect). Moreover to predict how much lower temperatures will be inside your house you need to know how much heat is currently transferred from the roof, and how much from other sources (e.g. windows, doors, cracks).
Please be aware that grass roofs can add a lot of weight, especially when it rains and lots of water is retained. For grass-covered roofs you need to take an extra 100-150 kg/m2 into account (and for sedum-covered roofs 40-80 kg/m2), so make sure your roof can handle this before you start.