My roomate just decided to move out this weekend. We have a lease through August. Can I demand rent from her? Will the landlord serve her a notice if I pay my half only?

Roomates are considered a single entity in terms of a lease, so you are both equally liable for rent and lease. However, in reverse each party is liable for the rent in FULL whether both live there or not. If you do not pay your rent in full, the landlord can serve a 3-day notice. You can not just pay your one half of rent and say the other person moved out.

Your options are to cough up the full month's rent and/or quickly find a new roomate. If the vacating roommate is on the lease you could take them to court for damages (their unpaid portion that you end up paying), but the effort may not be worth the end result.

Any occupancy changes usually will have to be agreed upon by your landlord. Most leases prohibit subletting (which you would be doing by bringing in a new roommate). Inform them of the situation, do whatever you can to make good on the rent. Your landlord is not obligated to help you with this situation, though. Your landlord may cooperate with an early move-out if they can show and re-rent the unit quickly so if they offer this as an option, you may be wise to accept.