Definitely don’t put anything in your name if the bill is link to other tenants that you do not live with. You will be liable for that bill every month. If your landlord has one giant bill for all his tenants, chances are he’s probably renting illegally. Check to see if he has an updated rental license and how many units the property is zoned for on this site: https://eclipse.phila.gov/phillylmsprod/pub/lms/Default.aspx?PossePresentation=BusinessLicenseSearchByLocRent&IconName=Contract_Mag.png

If I were you I would put your rent into escrow until you have heat, even ask for a discount since you have no heat…