Considering Acheta domecticus flour’s growing importance and utilization as an ingredient in many food products, research on its storage is essential. The objective of this study was to determine the chemical and nutritional fat profile of house cricket (Acheta domesticus) flour during storage for 12 months under different storage temperatures (−18 °C, +4 °C, and +20 °C in two variants, with and without access to light). Insect flour was studied using Fourier-transform infrared spectroscopy (FTIR). The fatty acids content was determined, and dietary indicators were calculated. The acid value, peroxide value, and anisidine value were also determined, and differential scanning calorimetry was performed. The results obtained from spectroscopic analysis of Acheta domesticus flour were consistent with the biochemical data. During the 12-month period of flour storage, the storage temperature significantly influenced the percentage composition of identified groups of fatty acids and the values of all presented ratios and dietary indices. During storage at refrigerated temperatures (−18 °C and +4 °C), no changes were observed in the fatty acid content and dietary indices, indicating that refrigerated temperatures provide oxidative stability to flour during 12 months of storage. Samples stored at 20 °C had higher acid values (AV), peroxide values (PV), and anisidine values (p-AV) compared to samples stored at lower temperatures (4 °C and −18 °C). Simultaneously, an increase in SFA and MUFA, as well as a decrease in PUFA and UFA, was noted in samples stored at room temperature. Storing cricket flour at lower temperatures when the storage period will be more than 12 months is essential to restrict the occurrence of fat oxidation. Elevated temperatures and exposure to light have a notable effect in hastening oxidation mechanisms, reducing thermal resilience, and inducing more pronounced alterations in the quality of fats.