This type of steel is known by several different names: one-pass, skin-pass, hot rolled pickled temper rolled sheet, etc. Here are three pages from US Steel that show the tolerances for (1) hot rolled pickled temper rolled sheet with (2) hot rolled pickled non-temper rolled sheet and with (3) cold rolled sheet:
Keep in mind that there is also only a narrow window of thickness where a steel company will offer the same grade as either hot rolled or cold rolled. This is likely only between 1.8 and 2.5mm.
Thin-gauge hot roll limits the tons per hour of the steel mill, so don't be surprised if there is a "gauge extra" in the pricing.
Thick gauge cold roll requires different hot band processing so that the cold mill can still achieve the approx 70% thickness reduction to produce the desired properties.
Bottom line: it isn't a guarantee that hot rolled steel will be less expensive than cold rolled steel across the board, or that it is even an option for your grade/coating/gauge of choice
-Danny