Should your big toe hurt in climbing shoes?
It is completely normal for your toes to feel sore in climbing shoes, especially if you are a beginner, and you are not used to them, or if you have downsized your shoes a lot. Everyone can have different toes that hurt, the big toe is a quite common one.
Though some amount of pain is considered normal, if you have throbbing pain that lasts even after you take your shoes off, then you may either have shoes that are too small and tight, or the shoes just don't fit the shape of your feet.
Finding the right climbing shoe that actually fits your feet is crucial. Not only to find the right size, but also the right shape, one that will fit your feet. This can really help ease most of the pain and discomfort that comes with climbing shoes.
It sounds like either the shoes are not the right shape or fit, or there is a pressure point on your big toe. Or maybe you put too much weight on your big toe rather than spreading the weight across all of your toes?