  The $\mathrm{SU}(4)$ Heisenberg model on the honeycomb lattice is expected to
host a quantum spin-orbital liquid at low temperature with an astonishing
candidate material, $\alpha$-ZrCl$_3$. We employed the canonical thermal pure
quantum state method to investigate the finite-temperature phase of this model.
Exploiting the full symmetry of $\mathrm{SU}(4)$, the calculation up to a
24-site cluster, which is equivalent to 48 sites in the spin-1/2 language, is
possible. This state-of-the-art computation with large-scale parallelization
enables us to capture the thermodynamic properties of the $\mathrm{SU}(4)$
Heisenberg model on the honeycomb lattice. In particular, the specific heat
shows a characteristic peak-and-shoulder structure, which should be related to
the nature of the low-temperature quantum spin-orbital liquid phase. We also
discuss what can be concluded from the assumption that the ground state is
gapped and symmetric in view of the generalized Lieb-Schultz-Mattis theorem.
