# If a key is used to encrypt that same key, is the computational cost to brute force it decreased?

If a key $$d$$ is used to encrypt a message $$m$$ that is identical to the key $$d$$, resulting in a cipher $$c$$

$$c = \text{Encrypt}(m, d) = \text{Encrypt}(d, d)$$

is the computational cost to brute force $$d$$ lower than if the key $$d \ne m$$ ?

In the IND-CPA (or even IND-CCA) security game, the adversary is not given the secret key $$d$$ (obviously) and hence cannot ask for $$\mathrm{Enc}(d,d)$$.
In other words: an encryption scheme where $$\mathrm{Enc}(d,d) = d$$ for all keys $$d$$ can still be IND-CPA(/CCA) secure (but they'd be completely useless for your scenario).