The Arrow of Time as the Sign of the Hubble Parameter

I argue that in Friedmann-Lemaitre-Robertson-Walker (FLRW) cosmology with ordinary matter obeying the null energy condition (NEC), the thermodynamic arrow of time aligns with the cosmological arrow identified by the sign of the Hubble parameter H = a_dot/a. Using apparent-horizon thermodynamics, the horizon entropy S_hor = pi/(G H^2) is monotone non-decreasing exactly when H > 0 and non-increasing when H < 0. Assuming a generalized second law (GSL) for the sum S_gen = S_matter + S_hor, the direction of non-decreasing S_gen coincides with expansion (H > 0). Edge cases (de Sitter, phantom w < -1, and bounces) are discussed.