Hello, tbh, this was my first time, too. Prior to this, I had only heard of New York cheesecake.
I asked because the cake I bought was cold, like it just came out of the fridge. I asked the clerk, and she said it was how it's supposed to be. I was a bit skeptical so I went home and googled, and it said new york cheesecake was supposed to be chilled, which made me think that a Basque cheesecake should be hot. But I really don't know, because like I said, it was my first time. But thanks for your reply anyway.