My tenant wants to move out 6 months before the lease ends. If I advertise the unit for rent and I am unable to find a new tenant for the next month, can I require the tenant to pay rent for that month?

When a tenant breaks a lease, the landlord has an obligation to mitigate their losses and try to re-rent the unit as soon as possible.  Landlords can file an L10 application to collect rent arrears and/or compensation that you believe the former tenant owes you with the LTB within one year from the date the tenant moved out of the rental unit. However, to file this application with the LTB, you will have to find out their new residential address to be able to serve them unless you have consent to email service in the lease agreement and can prove that the email came to the attention of the former tenant.