Can my Landlord as for Prorated Rent after the fact

[Heedy Williams]

My landlord asked me to sign the lease early on the 20th because she was going out of town and would not be back by the first.  On my year lease she changed the dates are 12/20/17 - 01/01/18 but kept the part where it says rent is due at the first of every month. I moved in on January 5th I was out of town prior to that. For the first 4 months I paid rent on the 20th as requested by the landlord.

I got a new job and new pay schedule and asked if I could start paying on 1st of the month. Landlord said that was fine, then days later changes her mind and says rent needs to be paid on 20th and I also now owe her prorated rent for moving in early on the 20th. Even though it was her idea to sign lease early and I did not in move in early, I have email correspondence between us to prove this.

I also asked specially about a prorate and if it will effected the rent or cost more when i signed and the landlord said no. Now the landlord's story has changed even though I said I will go back to paying on the 20th to avoid drama the landlord still wants the prorate.

Can my landlord do this ?