Yes it’s more expensive but as to how much you’ll need to ask the builder. It does add a lot more value to the house than a sloped extension and does look more cohesive. Extensions are always a game of compromise. You could add the pitched roof, knowing that when your quotes come back higher than expected it is one place that you can cut back.

the short answer could be yes but you need a quote from your builder to be confident about how much more and the internal raked ceilings also add another extra cost ... a gable roof is much nicer to look at but costs more to connect to the old roof and that's why "links" are often used to avoid changing the original roof and to allow courtyard area between the old and new but your plan suggests that the extension floor and roof heights are built at the same height as the original house and i'm wondering why your draftsperson hasn't also suggested a few steps down between the original and the extension with lower ceiling and slab on ground with different roof angle and scissor trusses to allow for the raked ceiling

Get your draftsman to draw up both concepts, and get your builder to quote on both roof concepts. This will then point you in the correct budget decision. But aesthetically what would look better internally? You will only do this once with this home. Important to get it right and on a budget that is manageable financially.