Which heating/cooling technology is the most sensible?

The question remains whether an air-water heat pump with a photovoltaic system makes sense.

These are two independent questions that should not be connected.
1. Is an air-water heat pump sensible? ==> Answer yes
2. Is photovoltaic sensible? ==> Answer yes

BUT: A photovoltaic system also makes sense with gas or oil (in existing buildings). Furthermore, gas alone is also sensible (for EnEV normally with solar thermal) without photovoltaic.

These are two independent questions that should not be connected. 1. Is an air-to-water heat pump sensible? ==> Answer yes 2. Is photovoltaics sensible? ==> Answer yes BUT: Photovoltaics also makes sense with gas or oil (in existing buildings). Furthermore, gas alone is also sensible (usually with solar thermal for EvEV) without photovoltaics.

Isn't it more likely that the running costs of an air-to-water heat pump + photovoltaic system are higher than with a gas boiler and a photovoltaic system?

Isn't it rather the case that the running costs of an air-water heat pump + photovoltaic system are likely to be higher than with a gas boiler and photovoltaic system?

Why? What did you calculate? Don't always fixate on the combination with the photovoltaic system. It doesn't change your heating costs.

Why? What did you calculate?

Don’t always get stuck on the combo with the photovoltaic system. It doesn’t change your heating costs.

I know they don’t go down. It’s about the fact that gas boiler = gas and air-to-water heat pump = electricity. Electricity is much more expensive than gas. But yes, you’re right. I need to make a calculation and compare.

It’s about the gas boiler = gas and air-water heat pump = electricity

So? You are comparing apples and oranges.
If you have a certain heating energy consumption, you can cover it with different energy sources.
For this, you have the COP with heat pumps, but on the other hand, possibly high connection costs for the gas line. You really need to know how much heating energy your house requires (not to be confused with the vague stuff from the energy certificate) and can then decide which "machine" produces the required energy.

I know they don't go down. It's about the gas boiler = gas and air-water heat pump = electricity. Electricity is much more expensive than gas. But yes, you're right. I have to make a calculation and compare.

Sort of, electricity is more expensive, but you don't have to draw the required heating demand 1:1 from electricity. With a heat pump, you have the annual performance factor as a guideline. So you only need a certain portion of electricity to generate sufficient heating output. With gas, you basically put in the same amount of gas as the heating output you need. With a heat pump, it depends on an optimized and well-dimensioned heating surface design to save heating costs. With gas, it's less of a concern.