In this paper, we report on the influence of molecular beam epitaxial (MBE) growth temperature on the spectral response of the long-wavelength infrared radiation (LWIR), three-stage thermoelectrically (TE) cooled (T = 210, 230 K) InAs/InAsSb type-II superlattice (T2SL)-based detectors grown on the GaSb/GaAs buffer layers/substrates. Likewise, antimony (Sb) composition and the superlattice (SL) period could be used for spectral response selection. The presented results indicate that the growth temperature affects the 50% cut-off (λ50%cut-off) of the fabricated devices and could be used for operating wavelength tunning. Assuming constant Sb composition and T2SL period during MBE process, the growth temperature is presented to influence λ50%cut-off covering entire LWIR (e.g., temperature growth change within the range of 400–450 °C contributes to the λ50%cut-off ~ 11.6–8.3 μm estimated for operating temperature, T = 230 K). An increase in temperature growth makes a blueshift of the λ50%cut-off, and this is postulated to be a consequence of a modification of the SL interfaces. These results show an approach to the T2SL InAs/InAsSb deposition optimization by the growth temperature in terms of the spectral response, without influencing the T2SLs’ structural properties (Sb composition, SL period).