Battery temps
It quite usual when a small turbine is started, to see a huge charge current when the genny is switched on. With a 200A starter/generator, it can easily be well over 100A initially.
On large aircraft such as the L1011, the battery system had a pulsed charger with separate control, rather than the battery being charged directly from the regulated bus.
For the smaller aircraft with a HOT BATTERY arning, it's just that - a warning. The pilot is meant to follow the Emergency procedues section of the AFM to reduce loads and land ASAP.