Property rights
First, as far as property rights go, there's no significant difference between a lease, and month to month tenancy. As a matter of fact, as a long time landlord, I tend to favor month to month over leases. IOW, if it turns out I want the tenant out, I don't have to screw around with lease stuff, and just give notice.
Regardless, a rental agreement or a lease grants "ownership" rights to the tenant. A landlord could put restrictions in the agreement, and all parties can agree, which may be legal depending on the restriction. But he can't make stuff up after the fact. Also, if a contract has illegal restrictions, it's "voidable" and not really legal.
Next, things about real estate MUST BE IN WRITING, to be legal and enforceable. For example, a landlord might verbally say I don't want you keeping guns here, but, if it's not in writing, it's just hot BS odored air.
A tenant has the right of 'quiet enjoyment' which means the landlord can't come in and inspect the tenants underwear for racing stripes any ole time he feels like it. IOW, entries are subject to prior notice. Now here's the exception: Entry can be made anytime to protect the actual landowner's interest in the property or for safety. This situation is an exception for emergencies like water/gas leaks and other things. Things that could damage the property or immediately affect safety of others or tenants. IOW, it has to be a real emergency.
The safety entrance exclusion would not apply to objectionable things, private fetishes, religions, whatever, or guns, even if prohibited in the agreement.
IOW, even if you SIGNED a rental agreement restriction against guns, the landlord can't come in to look for them. Now, if guns were seen by a maintenance guy fixing your AC, it could then become an issue. However, just because the AC guy saw a gun, it still DOES NOT allow the landlord to enter and search for them. He could ask you to vacate, but he can't come in.
Leases and rental agreements are fairly standard and you can find them on the web or in stationary stores or whatever. One thing in common, there is generally NO mention of guns in them, I've never seen one.