What about OpenTopoMap? Is it also third party project and refreshes for a very long time?

If you click "About" and then look under "Datenstand" it will tell you the date that OpenTopoMap was last updated.

Styles such as OpenCycleMap and OpenTopoMap typically have a more complex rendering chain, often including elevation data and complex route processing, than the fairly simple Standard (osm-carto) style on openstreetmap.org. They are also intended for third-party users such as cyclists and hikers, rather than for OSM mappers, so the authors probably consider that immediate updates are less important to their users.

If you want near-immediate updates then the standard style is always going to be your best bet.