What are (if any) the consequences to a Christian publicly denouncing Christ in order to save their lives?

Back during the early persecution of Christians in the Roman Empire, this was quite a big controversy. The Christian community was split between offering prayers to the emperor (and thus undermining their Christianity) and refusing to do so, which often meant a death sentence. Those that offered prayers felt it was no big deal since they were coerced, and believed that this was no big deal because in their hearts they were still fully Christian.