Cooling in summer with air-to-water heat pump, underfloor heating and/or ventilation system?

We have an air heat pump with reversible operation for heating and cooling. However, the manufacturer requires a connection between the outdoor and indoor units with suitable refrigerant lines. This would cost us about 1000 EUR. We have now decided on green roofing. It costs a bit more initially but less in maintenance and operation. Additionally, there are municipal subsidies, the sealed surface area is smaller, and temperature differences of up to 7 degrees Celsius are said to be achievable.

Just google "BERLIN Senatsverwaltung für Stadtentwicklung (2010)" in their brochure "Konzepte der Regenwasserbewirtschaftung. Gebäudebegrünung, Gebäudekühlung. Leitfaden für Planung, Bau, Betrieb und Wartung." It is further explained there that the evaporation of 1 m³ of water generates an evaporative cooling effect of 680 kWh,... The annual evaporation performance of extensive green roofs is based on the retention of 60 - 80% of the annual precipitation, which evaporates with an energy consumption of 2670 kJ per m². The effect of evaporative cooling can be further improved by watering the roof. As a rule of thumb, about 3-5 mm per square meter can evaporate per day. This means that with only 3 mm evaporation performance, 300 liters of water evaporate on 100 square meters of roof or facade area. This corresponds to an energy release of 200 kWh/day. If this cooling were generated by typical air conditioning systems, it would cost about €20.00/day at today's average electricity price.

Then you need an irrigation system. It's so dry that there is nothing to evaporate.

How about the statics? Such a roof weighs quite a bit.
How is the maintenance?
What about invasive growth? When I look at what grows on my stone desert in a year. The oat harvest was amazingly good.
How is it in winter? Does it cool as well?
Does it increase the snow load?

I don't think the idea of green roofs is bad. We once thought about it at the very beginning for the garage.

I am cautious with green roofs... Considering what roots are capable of lifting and breaking through, I actually don't want that on the roof. No matter how many times special coverings are called "root protection." We had a shoot that broke through a 100mm thick asphalt layer from below over the years. They can tell me a lot about root protection, etc.

Under asphalt, roots grow really well. Rootbarrier, I believe, gives a 25-year warranty.

I'll jump in here. Unfortunately, we have to green our flat roof at least extensively. Now the question for me is whether I should still invest extra money to be able to run the heat pump reversibly when a green roof supposedly provides so much cooling. Planned is a brine-water heat pump with trench collector.

Especially with a brine-water heat pump, I would take it since it is virtually free to operate, meaning you only need the electricity for the pump, the cold water is free of charge. The roof should provide significantly more than the ground cooling; I read about roughly a 2°C temperature delta that can be achieved with optimal passive cooling. In a summer like this one, those extra 2°C would probably be worth a few bucks to me.