Can an employer cut your pay as punishment?

Yes, an employer can generally cut pay as punishment. Reductions in pay rate, both temporarily or permanently, are legal, so long as your employer is not now paying you less than the federal minimum wage set.

You can decide to leave a job if you do not accept your pay cut, but keep in mind that this would be considered as you quitting your position, and you would not receive severance. In fact, there are companies that decrease pay in hopes that people leave and they do not need to pay severance pay by firing these individuals.