We investigated recovery in $Au/YBa_2Cu_3O_7$ (YBCO) thin film meander lines on sapphire substrates. The meander lines were fabricated by patterning YBCO films coated with gold layers. The lines were subjected to simulated AC fault current and then small current was applied for recovery measurements. The samples were immersed in liquid nitrogen during the experiment. After the fault, the resistance decreased linearly, first slowly and then fast to zero. The initial slow decrease was due to the decrease of the meander line temperature, whereas the fast decrease was originated from the transition from the normal state to the superconducting state. The recovery speed depended on the size of samples, and was faster in the smaller samples during the whole period of recovery. The experimental results were analyzed quantitatively with the concept of heat transfer within the sample and to the surrounding liquid nitrogen. A heat balance equation was solved for the initial phase of recovery, and an expression for the time dependence of resistance was obtained. The result agreed with data well.