In Verlinde's entropic force scenario of gravity, Newton's laws and Einstein equations can be obtained from the first principles and general assumptions. However, the equipartition law of energy is invalid at very low temperatures. We show clearly that the threshold of the equipartition law of energy is related with horizon of the universe. Thus, a one-dimensional Debye (ODD) model in the direction of radius of the modified entropic force (MEF) may be suitable in description of the accelerated expanding universe. We present a Friedmann cosmic dynamical model in the ODD-MEF framework. We examine carefully constraints on the ODD-MEF model from the Union2 compilation of the Supernova Cosmology Project (SCP) collaboration, the data from the observation of the large-scale structure (LSS) and the cosmic microwave background (CMB), i.e. SNe Ia+LSS+CMB. The combined numerical analysis gives the best-fit value of the model parameters ζ ≃ 10-9 and Ωm0 = 0.224, with χ2min = 591.156. The corresponding age of the universe agrees with the result of D. Spergel et al. [J.M. Bardeen, B. Carter, and S.W. Hawking, Commun. Math. Phys. 31 (1973) 161] at 95% confidence level. The numerical result also yields an accelerated expanding universe without invoking any kind of dark energy. Taking ζ(≡ 2πωD/H0) as a running parameter associated with the structure scale r, we obtain a possible unified scenario of the asymptotic flatness of the radial velocity dispersion of spiral galaxies, the accelerated expanding universe and the Pioneer 10/11 anomaly in the entropic force framework of Verlinde.