In current usage, penance is associated with spiritual practice as a form of self-imposed punishment:
penance noun: The performance of some act of self-mortification or the undergoing of some penalty as an expression of sorrow for sin or wrongdoing.
When the word penance came into English from Anglo-Norman, it applied to secular punishments as well as penalties assigned by religious authorities.
In modern usage, punishment is the usual word associated with a penalty imposed by a secular authority, whereas penance retains the idea of a religiously assigned or self-imposed act of retribution.
Post-classical Latin agere paenitentiam translates as “to do penance,” and the verb “to do” remains the idiomatic convention in modern English. For example, “Though still a promising star, Lohan will have to do penance before she’s forgiven for this boring, unfunny feature.”
Recently I have noticed the unidiomatic “to pay penance,” chiefly in the context of sports:
Pro football continues paying penance for some players’ involvement with domestic violence.—Steve Inskeep, NPR.
Pay your penance, and all is forgiven—ESPN headline.
Fordham is still paying penance for the colossal mistake it made when it left the MAAC to join the Patriot League in 1990.—New York Daily News.
People “pay for” their sins,” but “do penance” for them.