  From black hole thermodynamics, the Bekenstein bound has been proposed as a
universal thermal entropy bound. It has been further generalized to an
entanglement entropy bound which is valid even in a quantum system. In a
quantumly entangled system, the non-negativity of the relative entropy leads to
the entanglement entropy bound. When the entanglement entropy bound is
saturated, a quantum system satisfies the thermodynamics-like law with an
appropriately defined entanglement temperature. We show that the saturation of
the entanglement entropy bound accounts for a universal feature of the
entanglement temperature proportional to the inverse of the system size. In
addition, we show that the deformed modular Hamiltonian under a global quench
also satisfies the generalized entanglement entropy boundary after introducing
a new quantity called the entanglement chemical potential.
