What is the ruling on air conditioning connected to electricity through unlawful means? And is it permissible to remain in a place air-conditioned in this manner?

It is not permissible to evade paying the price of electricity provided by the state, because that is consuming public funds unjustly.

However, if there is a genuine necessity to use electricity, such that its non-use would lead to loss of life or extreme hardship, and they are unable to pay for it, then they may use it without paying as long as they are incapacitated. In this case, the state is obligated to provide it for free if it can. Otherwise, the affluent are obligated to support these distressed individuals, as it is a collective obligation (fard kifaya) upon the wealthy to alleviate harm from a Muslim.

If they are not compelled to use air conditioners, then it is not permissible to evade paying the price of electricity, and you should advise them. If they do not accept, you are not obligated to leave the place, because you have a right to stay there, and their forbidden way of air-conditioning it does not negate your right.