The kiosk may have had a wooden roof, although I have no idea how anyone would know that. The imposing structure may have been the entrance to the island.
At first glance, it looks like the stone brackets above the tops of the columns is a later addition, it has a copmletely different style than the rest of the structure – which is columns with curtain walls.
Both books I have on temples say that this kiosk probably had a wooden roof, and I wonder if that roof was on top of the columns, or if it was on the squarish beams.