I'm not sure what advantage installing a heavier deep cycle battery would have, but I've read of lithium batteries being installed, mostly in the Abarth models. I don't know why some failed early, but it could have been the high under-hood temps in the ICE cars.
The lithium batteries don't like the heat and shouldn't be used when the temp is below freezing. I've been using an Antigravity lithium battery off and on for two years. They have a very low amp hour rating so you don't sit around playing the radio unless the motor is running. Charging one in below freezing temps cause a build up in the battery that shortens it's life. So running one in the winter is not wise. Mine got hot one summer night and shut down on me . I was within two miles from home so I just had a friend bring me my lead acid battery. Since then I've moved the lithium battery to behind the passenger seat. If it's hot the ac will be on in the car. Under the hood temps have been measured and reach 140 degrees. I don't have a turbo ,my car is a 13' Sport. The lithium battery drops 35 lbs. off the car. My battery leads from inside the car tie into the OEM battery clamps. So installing the lead acid battery is quick and easy.