This work carries out a detailed investigation on the effects of rest time on the discharge response and the parameters of the Thevenin's equivalent circuit model for a lead acid battery. Traditional methods for battery modeling require a long rest time before a discharging test so that a steady state is reached for the open circuit voltage. In a recent work, we developed an algebraic method for parameter identification of circuit models for batteries by applying discharging tests after variable and possibly very short rest time. This new method opens a door to the understanding of the effects of rest time on battery behavior, which may be used for better simulation, analysis and design of battery powered systems for improved battery efficiency and state of health. As we used the new method to extract circuit parameters after different rest times, we observed some unexpected results on the relationship between the rest time and circuit parameters. The initial voltages on the capacitors can be negative and becomes more negative as the rest time is increased. We also observed that the time constants increase with rest time. Relationships between rest time and other parameters are also reported in this paper.