Would detection distance be changed by temperature of the environment?
Yes, it changes. The output of the sensitivity of a pyroelectric infrared sensor changes due to the temperature difference between the background temperature and the object (people). For this reason, as the temperature difference becomes lower the change in the output also becomes lower in principle, and the detection range becomes shorter as a result.
There is no output of sensitivity when the background temperature is equivalent to the body temperature in principle. Since the surface temperature of a human body is uneven, the detector will never completely stop detecting in practical use.