The landlord cannot terminate the lease before the end of the current lease period (the month, on whatever day is the anniversary of starting the lease), since this was not a condition of the lease. You have been notified of this new term, so starting with the next month, it is part of the conditions constituting the lease, whether you sign or not. (However, ...
When they start giving legal advice
... unless your PA is a lawyer.
In a nutshell, legal advice has the following characteristics:
Requires legal knowledge, skill, education and judgment
Applies specific law to a particular set of circumstances
Affects someone's legal rights or responsibilities
Creates rights and responsibilities in the ...
Your remedy will likely depend on your ability to show that John is responsible for heating this space. Is there a contract which shows this? Is it a condo rule?
From there you may be able to formulate a theory which might show John's failure to perform.