Is it possible to alarm a pool cage???

you mean like a fence?

yah probably, depending on fence and how much you want to spend and where the nearest wire is that goes to the alarm panel...etc etc.

liability would be high so I wouldn't do it for a client but for my family I probably would.

You could threaten to throw a cat into it.

There are a number of ways to provide "partial" security to a pool cage. I emphasize partial because it's difficult to totally protect an outdoor environment without false alarms. There's always a compromise.

One method is to use wireless, outdoor motion detectors. There's a system called Voice Alert which uses from one to six wireless PIR (passive infrared) sensors and a dedicated receiver with voice annunciation (you record your own messages) and relay outputs (for connection to an existing alarm. The idea is to provide you with an audible warning but typically no siren if you're home and someone enters the lanai. When you're away you might want the main siren to also be triggered on entry.

Most folks recommend against sending a signal to a monitoring facility based on outdoor motion due to the higher incidence of false alarms from this type of detector. If you plan to do that, you might want to consult with local authorities first. There may be local regulations prohibiting it.

Other types of detection that can be made to work with a bit more effort include magnetic contacts on the screen doors, infrared beam detectors around the perimeter and even floating sensors to detect a splash. Note that the floating sensors, which are primarily used to protect against accidental drowning, are not UL listed and rarely meet state code requirements for the purpose.

Speaking of codes, Florida, California and several other states now require pool alarms on the doors and windows leading from the home to the pool area. This is intended to help avoid toddlers drowning. The detectors are wired to the doors and windows and set to give an audible warning inside the house if someone opens a door to the pool area. Some provide a 7-second delay so you can re-enter while others utilize a separate button mounted on the outside wall above the reach of a child, to temporarily disengage the sensor before opening the door. None of these pool alarm systems are 100% effective but they do help when used properly.

You can have all sorts of outdoor sensors. A magnetic contact for if a gate is opened. A light beam that if it is broken, it trips the alarm. A "trip wire" that if someone walks into it, an electrically conductive tab pulls out of a pressure device which then loses electrical contact and trips the alarm (or if wire is cut).

Then the military and prisons have all sorts of sensors including ground pressure sensors and motion sensors and wires running in fences (trip if cut or capacitance trip if touched), however these are prone to false alarms, but in these cases there are 24 hour armed guards who are patrolling the fence anyway, so not a big deal. Or they might have cameras and if an alarm triggers at a remote location, they can look via the cameras to see if it was the wind blowing a box into a fence or an intruder.

Big vicious dog.

Alarm can be used to trip a sensor on a cage door to contain the dog so pool can be used by family. Timer in the alarm system can be set to activate an automatic feeder. Pay the guy who cuts the lawn a few extra bucks to clean up the "droppings". If he doesn't do a good job, just leave the cage door unlatched.