Are writers who are classified as "Gothic writers" automatically part of the classic literary canon?
A particularly clear definition of the term “literary canon” is the one that describes it as referring
to those works in anthologies that have come to be considered standard or traditionally included in the classroom and published textbooks. In this sense, "the canon" denotes the entire body of literature traditionally thought to be suitable for admiration and study. [see link below]
In this sense, no particular work – Gothic or otherwise – is automatically part of the “literary canon.” According to the definition given above, only Gothic works that have struck a great many readers as particularly worthy of “admiration and study” are included in the literary canon. Mary Shelley’s novel Frankenstein would be one such work: it exists in numerous editions, it is very widely read and studied, it is discussed in practically any history of “Gothic” literature, and admiration of the work has endured for a very lengthy period. In all these ways, Frankenstein is included in the “literary canon” in a way that the Twilight novels (for instance) are not. It is possible that someday the Twilight novels will achieve the kind of stature and recognition and appreciation that Frankenstein has achieved. If they do, they will then be part of the canon. It is also conceivable that someday Frankenstein will lose its luster and no longer be widely read. If that ever happens, then Frankenstein will be a far less visible and important part of the canon than it is today.
In the broadest possible sense, every single work of literature that has ever been written might be considered part of a “literary canon,” but almost no one uses the term “literary canon” in that sense.
However, disputes about which works should and should not be part of the canon occur all the time, especially in the case of contemporary literature. Disputes are far less likely to occur about works that have been “canonized” in the past. For example, John Dryden’s poems are always likely to be considered part of the English literary canon because they were so highly regarded in their own time and in subsequent periods. Even if it should happen that someday almost nobody reads Dryden’s poetry, his works will probably never be considered non-canonical.