Who pays for the pool maintenance depends on your lease agreement-they vary depending upon the home owners and or tenants wishes. Some landlords charge a little extra to pay for expenses such as monthly pool maintenance, & landscaping while others allow the tenant to maintain those things. Typically the landlord pays for mechanical repairs but not for added upkeep due to neglect by the tenant.
Again, you need to read your lease agreement. If the landlord stated that will maintain the pool, then they they must do it. If it doesn't say either way, it is likely your responsibility.
Best of Luck
Premier Real Estate