First off, let us not confuse this Eusebius with Eusebius of Caesarea of Church History fame. As papacies go, Eusebius' really did not have much time to come to fruition, lasting only about 4 months, most of which was spent in exile.
That said there is a very important result from his leadership.
Life pretty much sucked for Christians in the early part of the fourth century, and many renounced their Faith in order to protect their lives and those of their families (and so known as "apostates"). For some this was the serious sin: basically blasphemy against the Spirit - the one unforgivable sin according to Jesus. One had turned one's back on Christ, who said that he would turn his back on those who did that.
Eusebius maintained and reiterated the teaching established a century earlier that, with sufficient penance, those who apostatized could return to the fold (it seems that those who had wanted to come back demanded unconditional return with no questions asked). The ensuing fight got everyone exiled. Eusebius died during that exile and was brought back to Rome by his successor to a martyr's honor.
It shows again the struggle to understand sin and forgiveness. Sin can be forgiven but it does require an element of penance and true repentance. Forgiveness is given freely by the Father through the death of Jesus, but it requires a sincere heart and effort. There is no free ride.
“Teacher, what good must I do to gain eternal life?” He answered him, “Why do you ask me about the good? There is only One who is good. If you wish to enter into life, keep the commandments.” He asked him, “Which ones?” And Jesus replied, “You shall not kill; you shall not commit adultery; you shall not steal; you shall not bear false witness; honor your father and your mother; and you shall love your neighbor as yourself.” The young man said to him, “All of these I have observed. What do I still lack?” Jesus said to him, “If you wish to be perfect, go, sell what you have and give to the poor, and you will have treasure in heaven. Then come, follow me.” When the young man heard this statement, he went away sad, for he had many possessions.
-- Matthew 19:16-22