We investigate the thermal topology in a super-penumbral canopy by determining the threedimensional (3D) thermal structure of an active region. We derive the temperature stratifications in the active region by an inversion of the Ca ii IR line at 854.2 nm, assuming local thermal equilibrium (LTE). We trace the 3D topology of individual features located in the super-penumbral canopy, mainly radially oriented fibrils. We find that about half of the fibrils form short, arched, low-lying loops in the temperature cube. These closed loops connect from bright grains that are either in or close to the penumbra to the photosphere a few Mms away from the sunspot. They reach less than 1 Mm in height. The other half of the fibrils rise with distance from the sunspot until they leave the Ca ii IR formation height. Many of the fibrils show a central dark core and two lateral brightenings as seen in line-core intensity images. The corresponding velocity image shows fibrils that are as wide as the fibrils seen in intensity without a lateral substructure. Additionally, we study one example of exceptional brightness in more detail. It belongs to a different class of structures without prominent mass flows and with a 3D topology formed by two parallel, closed loops connecting patches of opposite polarity. We present evidence that the inverse Evershed flow into the sunspot in the lower chromosphere is the consequence of siphon flows along short loops that connect photospheric foot points. The dark-cored structure of the chromospheric fibrils cannot have an convective origin because of their location above regular granulation in an optically thin atmosphere. The dark core most likely results from an opacity difference between the central axis and the lateral edges caused by the significant flow speed along the fibrils.