  Over the last decade, the interest in the spin-$1/2$ Heisenberg
antiferromagnet (HAF) on the square-kagome (also called shuriken) lattice has
been growing as a model system of quantum magnetism with a quantum paramagnetic
ground state, flat-band physics near the saturation field, and quantum scars.
Here, we present large-scale numerical investigations of the specific heat
$C(T)$, the entropy $S(T)$ as well as the susceptibility $\chi(T)$ by means of
the finite-temperature Lanczos method for system sizes of
$N=18,24,30,36,42,48$, and $N=54$. We find that the specific heat exhibits a
low-temperature shoulder below the major maximum which can be attributed to
low-lying singlet excitations filling the singlet-triplet gap, which is
significantly larger than the singlet-singlet gap. This observation is further
supported by the behavior of the entropy $S(T)$, where a change in the
curvature is present just at about $T/J=0.2$, the same temperature where the
shoulder in $C$ sets in. For the susceptibility the low-lying singlet
excitations are irrelevant, and the singlet-triplet gap leads to an
exponentially activated low-temperature behavior. The maximum in $\chi(T)$ is
found at a pretty low temperature $T_{\rm max}/J=0.146$ (for $N=42$) compared
to $T_{\rm max}/J=0.935$ for the unfrustrated square-lattice HAF signaling the
crucial role of frustration also for the susceptibility. We find a striking
similarity of our square-kagome data with the corresponding ones for the kagome
HAF down to very low $T$. The magnetization process featuring plateaus and
jumps and the field dependence of the specific heat that exhibits
characteristic peculiarities attributed to the existence of a flat one-magnon
band are as well discussed.
