The primary objective of this study is to examine the melting rate of phase change material (PCM), the natural convection characteristics, and the bottom temperature distributions of the PCM-based cylindrical heat sinks with horizontal fins. A two-dimensional numerical analysis was performed, and the effect of various horizontal fin positions on the cooling performance of the heat sink was investigated. Compared to the cases without horizontal fins, adding horizontal fins increased the surface area for heat conduction and increased heat absorption into the PCM, resulting in a lower bottom temperature rise. Horizontal fins impede natural convection, but the effect of reducing the rate of increase in the bottom temperature of the heat sink was more significant due to the increased surface area. The set point temperature (SPT) arrival time increased by up to 44% in the case with horizontal fins compared to the case without horizontal fins. Depending on the arrangement of the horizontal fins, SPT arrival time showed a difference of up to 11%. The results showed that the position of the horizontal fins inside the cylindrical heat sink is a significant geometrical parameter that affects the PCM melting rate and the bottom temperature of the heat sink.