We investigate a thermodynamic arrow associated with quantum projective measurements in terms of the Jensen-Shannon divergence between the probability distribution of energy change caused by the measurements and its time reversal counterpart. Two physical quantities appear to govern the asymptotic values of the time asymmetry. For an initial equilibrium ensemble prepared at a high temperature, the energy fluctuations determine the convergence of the time asymmetry approaching zero. At low temperatures, finite survival probability of the ground state limits the time asymmetry to be less than $ln 2$. We illustrate our results for a concrete system and discuss the fixed point of the time asymmetry in the limit of infinitely repeated projections.
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