custom.py

# /collections/{collection_id}/time/{date}/tiles/{tileMatrixSetId}/{x}/{y}/{z}.png

def DatasetPathParams(
    collection_id: Annotated[str, Path(description="collection_id")],
    time_dimension: Annotated[str, Path(description="date", alias="date")],
) -> str:
    """"Here you can use collection_id and time_dimension to get the COG url."""
    ...


app = FastAPI()

tiler = TilerFactory(
    path_dependency=DatasetPathParams,
    router_prefix="/collections/{collection_id}/time/{date}",
)

app.include_router(
    tiler.router,
    prefix="/collections/{collection_id}/time/{date}",
)