  $Context$. The thermal structure of the penumbra below its visible surface
(i.e., $\tau_5 \ge 1$) 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: $\tau_5 \in [1,3]$ ($\approx
70-80$ km below the visible continuum-forming layer). $Methods$. We analyzed
spectropolarimetric data (i.e., Stokes profiles) in three Fe \textsc{i} lines
located at 1565 nm observed with the GRIS instrument attached to the 1.5-meter
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 $T(\tau_5)$. $Results$. 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 $\tau_5 \ge 1$ 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. 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.
