Architectural planning costs 50% more expensive than agreed.

A budget does not have to be included in the architect's contract. There are already clear rulings on this. An architect is always obliged to inquire about the client's financial possibilities and to plan accordingly. If the costs are significantly exceeded, the planning is flawed, but if you have accepted this, there is naturally a problem.

Oh dear! I see a bad ending if you don't sit down with the architect and resolve the problem.

If you no longer want him, then agree on an amount that the HOAI calculator shows you. Here you can enter how much of each [LP] is fulfilled...
You can't avoid [Leistungsphase 1] and [2], and [Leistungsphase 3] proportionally?
The lawyer will also take money from you again, and the settlement amount will not be less.
If there is still trust, then now give up the "sheep attitude" and hold the guy responsible to "make improvements" as he has not completed the contract... if you have demanded improvements a few times without him doing anything, he holds the black Peter...
Good luck!

I see it is about time that the OP answers my follow-up question

How is the commission to the architect defined in the contract?

once.

A budget does not have to be included in the architect’s contract.

Correct, the budget belongs to the basic evaluation – but also statements from the clients. If they say they expect one hundred seventy square meters, but it "becomes" two hundred forty, then the architect hasn’t had enough target water. Even with a budget "carte blanche" it would then be necessary to obtain explicit approval for the increased size.

If you don’t want him anymore then agree on a sum that the HOAI calculator outputs to you. Here you can enter how much of the respective service phase has been fulfilled...
You can’t get around service phases 1 and 2 and service phase 3 proportionally?
The lawyer will also charge you again and the settlement sum will not be less.

Without a lawyer, however, a mistake is quickly made here – for example, following your advice: because you cannot simply unilaterally terminate the architect’s contract. And the HOAI calculator would probably output too high a sum in this case, since without a valid complaint the invoice would remain oriented to the too large house.

and hold the guy responsible to "make corrections" since he has not implemented the commission... if you have requested corrections a few times without him doing anything, then he holds the black Peter...

This brings us back to my follow-up question, what exactly is agreed upon in the architect’s contract. On this basis one should request corrections (and when saving at the wrong end by doing this without a lawyer and regarding form and deadline) or offer a termination (for which, again, the lawyer will better know how poor performance would be appropriately taken into account).

A budget does not have to be included in the architect contract. There are already clear rulings on this. An architect is always obliged to inquire about the financial possibilities of the client and to plan accordingly. If he significantly exceeds it, the planning is faulty, but if you accepted it, of course there is a problem.

um, no! We had a contract with a written budget. Cost overruns at least 80%, rather significantly above that. Everything was immediately complained about. Nevertheless, no chance, including lawyer fees about €20,000 thrown away and completely started over. Chances of a lawsuit are low because the term "binding cost ceiling" is not stated in the contract, only a sum designated as budget was fixed.

Wow, that is really crazy and makes me doubt the rule of law. It always depends on the individual circumstances. I referred to a ruling by the Bundesgerichtshof from 2013. It states: "the architect is basically obliged, already within the framework of the so-called basic investigation, to define the economic framework for a construction project with the client and to take their cost expectations into account. These cost expectations expressed to the architect are binding in the sense that they – subject to a subsequent change – determine the planning framework and in any case regularly become part of the contract if the architect does not object to them. According to the decision of the Bundesgerichtshof, such cost expectations are also to be considered even if they do not contain an exact upper limit of the construction sum, but only indications of the approximate construction sum, with which a cost framework is defined."

We had a contract with a written budget. Cost overrun at least 80%, rather significantly more. Everything complained about immediately. Still no chance, including attorney fees about €20,000 thrown away and started completely anew. Chances of a lawsuit are low because the term "binding cost cap" is not stated in the contract, only a sum fixed as the budget.

Wow, that's really intense and makes me doubt the rule of law.

I only doubt the lawyer. Apparently, with some lawyers, you shouldn't go there on Wednesdays around noon if the golf course is already calling. Seeing low chances of a lawsuit with an 80% budget overrun, I read that as a clear sign of laziness.