I can’t see a medical reason why you can’t go to the gym or do cardio after having double chin injections.
That being said – I wouldn’t personally recommend going for a workout after treatment. The area under the chin can feel a little sore and swollen and I think it would be better to go home, put some ice on the area and put your feet up.
I think going to the gym or for a workout the day after would be better. Enjoy a rest day!