I don't know the hardware he has but my brother in Florida wanted to do the same thing.
He called his security system vender and they upgraded his system to include a cellular phone with a remote antenna hidden in the attic and tied it all in with the same UPS his alarm system used.
As mentioned by Pat, the perps in my brother's area like to cut the cable, phone, and power connections prior to breaking in. Seems his local area requires an external power cut off as part of the electrical meter base making killing the power quite easy.
So now even with the power off, no cable or landline working, his alarm system along with the following options continues to work, high/low temperature, fire alarm, flooding, and combined cable + AC power failure, and panic buttons.