  We demonstrate the existence of prethermal discrete time crystals whose
sub-harmonic response is entirely localized to zero-dimensional corner modes.
Within the exponentially long prethermal regime, we show that the robustness of
these corner modes arises from two related, yet distinct mechanisms: the
presence of a higher-order symmetry-protected topological phase in the
effective Hamiltonian, or the emergence of a dynamical constraint that prevents
the decay of the corner mode. While the first mechanism ensures the stability
of the sub-harmonic response throughout the entirety of the prethermal regime,
it is restricted to initial states in the ground state manifold of the
effective Hamiltonian. By contrast, the second mechanism enables the
observation of the prethermal time-crystalline order for arbitrary initial
states, albeit with a time scale that is not only determined by the frequency
of the drive, but also the relative energy scale across the system's
sublattices. We characterize these two mechanisms by simulating the dynamics of
a periodically driven two-dimensional spin model, and discuss natural
extensions of our model to all other dimensions.
