  In BCS superconductors, the superfluid stiffness is virtually constant at low
temperature and only slightly affected by the exponentially low density of
thermal quasiparticles. Here, we present an experimental and theoretical study
on the temperature dependence of superfluid stiffness $\Theta\left(T\right)$ in
a strongly disordered pseudo-gaped superconductor, amorphous $\text{InO}_{x}$,
which exhibits non-BCS behavior. Experimentally, we report an unusual power-law
suppression of the superfluid stiffness $\delta\Theta\left(T\right)\propto
T^{b}$ at $T\ll T_{c}$, with $b\sim1.6$, which we measured via the frequency
shift of microwave resonators. Theoretically, by combining analytical and
numerical methods to a model of a disordered superconductor with pseudogap and
spatial inhomogeneities of the superconducting order parameter, we found a
qualitatively similar low-temperature power-law behavior with exponent
$b\sim1.6-3$ being disorder-dependent. This power-law suppression of the
superfluid density occurs mainly due to the broad distribution of the
superconducting order parameter that is known to exist in such superconductors
[arXiv:1012.3630], even moderately far from the superconductor-insulator
transition. The presence of the power-law dependence
$\delta\Theta\left(T\right)\propto T^{b}$ at low $T\ll T_{c}$ demonstrates the
existence of low-energy collective excitations; in turn, it implies the
presence of a new channel of dissipation in inhomogeneous superconductors
caused by sub-gap excitations that are not quasiparticles. Our findings have
implications for the use of strongly disordered superconductors as
superinductance in quantum circuits.
