The second law of thermodynamics in presence of gravity is known to hold at small scales, as in the case of black holes and self-gravitating radiation spheres. Using the Friedmann-Lema^{i}tre-Robertson-Walker metric and the history of the Hubble factor, we argue that this law also holds at cosmological scales. Then, we study the connection between the deceleration parameter and the spatial curvature parameter of the metric and set limits on it valid for any homogeneous and isotropic cosmological model. Likewise, we devise strategies to determine the sign of the index of the spatial curvature. Finally assuming the lambda cold dark matter model correct, we find that the acceleration of the cosmic expansion is increasing today.