Funny, I went the other way round in the Bay Area. PG&E bills were so high so it was the choice of putting on a jacket or paying $1k extra over the winter months. And my reasoning was "I can afford a jacket".
I bought some fleece sweatpants for $40 and the amount I was able to keep my home unheated was totally fine and I made that money back within that month.
Also the cost of apartment-friendly insulation paid for itself immediately as well (which wouldn't apply to most decent apartments).
And if your landlord is balking at including heat in rent, there's a decent chance it's because your bill will be outrageous because there's zero insulation, and as a tenant there's little you can do to fix that.