Enzymatic browning is a common limiting factor in the fruit industry because it causes significant losses through fresh product deterioration, affecting taste, flavor, and nutrition. Artocarpus odoratissimus, locally known as terap, is an exotic indigenous fruit to Borneo. This fruit remains underutilized due to its perishability, short shelf-life, and peel browning. Currently, no information has apparently been published on the browning mechanisms of A. odoratissimus. Thus, the present study aims to evaluate the degree of browning and enzymatic activities in relation to the phytochemical contents in A. odoratissimus during postharvest ripening. The experimental design consists of packaging (without packaging and with packaging) and storage temperatures (ambient at 25 °C, and cold storage at 10 °C), which were arranged in a randomized complete block design. Findings showed significantly higher weight loss in fruit stored at 25 °C on days 4 (T1) at 12.20 ± 0.19% and 8 (T5) at 11.09 ± 0.24%. The degree of browning was higher in the fruits stored with packaging at 25 °C, collected on day 4 at 0.48 ± 0.01 and day 8 at 0.51 ± 0.02, and consequently, the phenylalanine ammonia-lyase (671.00 ± 5.25 UE g−1 min−1) and polyphenol oxidase (670.00 ± 2.56 UE g−1 min−1) enzymatic activities were also higher. The lower enzymatic activities were recorded in the fruit stored at 10 °C without packaging, resulting in the least degree of browning. The fruit with the lowest enzymatic activities was stored at 10 °C without packaging, resulting in the least amount of browning. This trend is supported by lower total phenolic content (TPC) and is explained by a strong positive correlation between TPC and PAL (r = 0.927). Low-temperature storage was effective in reducing the effect of browning and deterioration on A. odoratissimus for up to 16 days. The results not only provided insights into the peel browning in A. odoratissimus but also guidance on controlling postharvest fruit browning.