My apartment lease says that the Internet has a telecom fee but there's no mention of how much the telecom fee is anywhere in the contract. In this case, am I obligated to pay this fee? Does it not invoke an issue where they can arbitrarily set the price? Does this fall under the category of lease addendum?
am I obligated to pay this fee?
Yes. You entered the lease despite been aware of the existence of that fee. That meets the contract law tenet that an agreement be entered knowingly and willfully.
Does it not invoke an issue where they can arbitrarily set the price?
If the landlord requires a fee that is unreasonably high, that would violate the contract law covenant of good faith and fair dealing. As such, that part would be unenforceable, meaning that the landlord may only charge a reasonable fee. Likewise, unless the lease clearly entitles the landlord to decide the telecom fee arbitrarily, the lease would fail to meet that extent of the aforementioned tenet of a contract being entered knowingly.
Does this fall under the category of lease addendum?
The form (or format) in which the requirement of telecom fee was mentioned --and agreed upon-- does not matter. The landlord only would need to prove that the tenant was --or should have been-- aware of that fee. Usually the landlord can prove that by showing/producing in court the lease with tenant's signature.