  Here we examine the role of the amorphous insulating substrate in the thermal
relaxation in thin NbN, InO$_x$, and Au/Ni films at temperatures above 5 K. The
studied samples are made up of metal bridges on an amorphous insulating layer
lying on or suspended above a crystalline substrate. Noise thermometry was used
to measure the electron temperature $T_e$ of the films as a function of Joule
power per unit of area $P_{2D}$. In all samples, we observe the dependence
$P_{2D}\propto T_e^n$ with the exponent $n\simeq 2$, which is inconsistent with
both electron-phonon coupling and Kapitza thermal resistance. In suspended
samples, the functional dependence of $P_{2D}(T_e)$ on the length of the
amorphous insulating layer is consistent with the linear $T$-dependence of the
thermal conductivity, which is related to lattice excitations (diffusons) for
the phonon mean free path smaller than the dominant phonon wavelength. Our
findings are important for understanding the operation of devices embedded in
amorphous dielectrics.
