Although I do not practice law in the State of Arizona, here are my initial impressions.  First, it appears that you are stating that the landlord's failure to fix the air conditioning is what you are claiming to be your breach of contract; however, if your landlord is not requiring you to pay your rent I am unclear as to what your potential damages are.  Nevertheless, putting aside the merits of your case, in any civil matter the general rule is that it is possible for an attorney to attach the defendant's assets (in this case a home) during the pendency of the lawsuit.  However, the attorney is usually required to demonstrate a likelihood of success with respect to the underlying matter prior to being able to attach any assets/put a lien on any property.  Therefore, I suggest that you consult with and/or retain an attorney that specializes in the area of landlord/tenant law to determine the most effective options available to you to protect your rights and otherwise resolve this matter favorably.