Sore lower back after a run. Should I stop?

No, but pay attn . A mildly sore low back is nothing to be concerned about. If it is severe, that's a different story. However a sore back is probably indicative of something going wrong. Most commonly, it is a weak core, poor form, or overtraining. One way to deal with it is to back of the volume & intensity of your running while adding in more core. Once it is better, ramp back up while maintaining the core.