That might be somewhat easy. If the animal has variation in the sound, such as length and pitch, then things might be tricky.
You could start it off by recording the sound of an animal that has very low variation in the sound. You would record this by creating an array(or many variables) and then recording the sound level every millisecond for 20-50 milliseconds, and putting them into the array(or puting them in their own variables.) You could turn this array(or your variables) into the threshold of the sound.
As for detecting a sound, you could set the Arduino to record the sound. You could then proceed to compare each millisecond of the sound to the threshold, and, if there is a low variation from the threshold, you could add a 1 to another array(I’ll explain this later). If there is high variation, you could add a 0 to the array. With the new aray, you can determine if the sound was made by the animal or not by confirming whether 5%-10% of the array is a 0. As for triggering the recorder, you could use the first three milliseconds of the threshold and perpetually compare them to the sound levels of the environment, triggering the recorder when the sound levels almost match the threshold.
In theory you could do that. I also have an idea for an advanced sound detector if the Arduino’s software permits it. Contact me if you would like me to explain my idea for the advanced detector.