© 2019 American Physical Society. The holographic principle can lead to cosmological scenarios, i.e., holographic equipartition models. In this model, an extra driving term (corresponding to a time-varying cosmological term) in cosmological equations depends on an associated entropy on the horizon of the Universe. The driving term is expected to be constrained by the second law of thermodynamics, as if the cosmological constant problem could be discussed from a thermodynamics viewpoint. In the present study, an arbitrary entropy on the horizon, SH, is assumed, extending a previous analysis based on particular entropies [Phys. Rev. D 96, 103507 (2017)PRVDAQ2470-001010.1103/PhysRevD.96.103507]. The arbitrary entropy is applied to the holographic equipartition model, in order to universally examine thermodynamic constraints on the driving term in a flat Friedmann-Robertson-Walker universe at late times. The second law of thermodynamics for the holographic equipartition model is found to constrain the upper limit of the driving term, even if the arbitrary entropy is assumed. The upper limit implies that the order of the driving term is likely consistent with the order of the cosmological constant measured by observations. An approximately equivalent upper limit can be obtained from the positivity of SH in the holographic equipartition model.