It is often needed in computational chemistry to compute free energy differences among multiple thermodynamic states. A common approach for this task is to build a perturbation graph connecting the states, and compute free energy differences on all edges of the graph. The free energy difference between any two states can then be obtained by integrating along a path connecting the two states. When the perturbation graph has cycles, there will be multiple paths connecting two states. Because free energy is a function of states, the free energy difference between any two states should be independent of the path chosen. Equivalently, the free energy around any cycle is zero, which we refer to as the cycle consistency condition. The cycle consistency condition, on the one hand, could help diagnose potential systematic problems in the calculation. For example, if the free energy around a cycle is significantly different from zero in practical calculations, it indicates that there might be systematic issues in sampling, such as the system being trapped in a meta-stable state. On the other hand, if there are no systematic sampling issues, the cycle consistency condition can be used to improve the accuracy of free energy estimates.