Context. The thermal structure of the penumbra below its visible surface (i.e., τ5 ≥ 1) has important implications for our present understanding of sunspots and their penumbrae: their brightness and energy transport, mode conversion of magneto-acoustic waves, sunspot seismology, and so forth. Aims: We aim at determining the thermal stratification in the layers immediately beneath the visible surface of the penumbra: τ5 ∈ [1,3] (≈70-80 km below the visible continuum-forming layer) Methods: We analyzed spectropolarimetric data (i.e., Stokes profiles) in three Fe i lines located at 1565 nm observed with the GRIS instrument attached to the 1.5-m solar telescope GREGOR. The data are corrected for the smearing effects of wide-angle scattered light and then subjected to an inversion code for the radiative transfer equation in order to retrieve, among others, the temperature as a function of optical depth T(τ5). Results: We find that the temperature gradient below the visible surface of the penumbra is smaller than in the quiet Sun. This implies that in the region τ5 ≥ 1 the penumbral temperature diverges from that of the quiet Sun. The same result is obtained when focusing only on the thermal structure below the surface of bright penumbral filaments. Conclusions: We interpret these results as evidence of a thick penumbra, whereby the magnetopause is not located near its visible surface. In addition, we find that the temperature gradient in bright penumbral filaments is lower than in granules. This can be explained in terms of the limited expansion of a hot upflow inside a penumbral filament relative to a granular upflow, as magnetic pressure and tension forces from the surrounding penumbral magnetic field hinder an expansion like this.