The unpredictable inherent error behavior of approximate computing introduces both new security threats and opportunities to design novel security primitives/strategies. This work proposes a methodology that exploits stochastic timing errors of a pipelined datapath caused by voltage scaling to design an optimized processor-based physical unclonable function (PUF) for approximate computing. To verify the effectiveness of this method, a pipelined arithmetic architecture is implemented at a 45 nm technology node, and voltage scaling is applied to extract PUF bits. With reduced supply voltage, harvested PUF bits show increased uniqueness. Moreover, proposed divergent delay path selection based on intermediary error behavior exhibits improved PUF uniqueness vs an unmodified datapath. A design optimization methodology is applied introducing new PUF metrics - gain (G) and performance power ratio (PPR). Using these metrics, the optimum scaled voltage range is identified for enhanced PUF performance. The optimized PUF shows maximum uniqueness of 49%, and reliability of 92% with a temperature range of -20${\circ }$C to 70${\circ }$C. Further, the proposed PUF with approximate computing achieves markedly improved G and PPR relative to the exact case. With better uniqueness, reliability, and low resource utilization, the proposed PUF methodology is highly suitable for securing approximate computing applications.