SuttaCentral
I am curious, could arahants be curious? And in what sense.
Could curiousity contain craving or be entirely free from craving? Does the Abhidhamma said anything about this?
When the Buddha asked ananda, what’s that noise? Can that be regarded as a form of curiosity?
In the suttas, sekhas, asekhas and even the Buddha himself are all occasionally reported as being curious about something or other.
A sotāpanna’s curiosity:
Then, while the Venerable Mahāmoggallāna was alone in seclusion, the following course of thought arose in him: “Which devas know: ‘I am a stream-enterer, no longer subject to rebirth in the lower world, fixed in destiny, heading for enlightenment’?”
Two arahantas’ curiosity:
Then, when the Venerable Sāriputta was alone in seclusion, the following course of thought arose in his mind: “What should a bhikkhu honor, respect, and dwell in dependence on so that he might abandon the unwholesome and develop the wholesome?”
Then it occurred to the Venerable Mahāmoggallāna: “Where now is the Blessed One dwelling at present?” With the divine eye, which is purified and surpasses the human, the Venerable Mahāmoggallāna saw the Blessed One sitting cross-legged in the air above that brahmā, having entered into meditation on the fire element.
The Venerable Sāriputta said this:
“Here, friends, when I was alone in seclusion, a reflection arose in my mind thus: ‘Is there anything in the world through the change and alteration of which sorrow, lamentation, pain, displeasure, and despair might arise in me?’ Then it occurred to me: ‘There is nothing in the world through the change and alteration of which sorrow, lamentation, pain, displeasure, and despair might arise in me.’”
An arahantī’s curiosity:
Then it occurred to the bhikkhunī Āḷavikā: “Now who is it that recited the verse—a human being or a nonhuman being?” Then it occurred to her: “This is Māra the Evil One, who has recited the verse desiring to arouse fear, trepidation, and terror in me, desiring to make me fall away from seclusion.”
A Buddha’s curiosity:
Then, when the Blessed One was alone in seclusion, a reflection arose in his mind thus: “Is it possible to exercise rulership righteously: without killing and without instigating others to kill, without confiscating and without instigating others to confiscate, without sorrowing and without causing sorrow?”
Then, while I was alone in seclusion, a course of thought arose in my mind thus: ‘It is painful to dwell without reverence and deference. Now what ascetic or brahmin can I honor, respect, and dwell in dependence on?’
Perhaps what an asekha is freed from is idle curiosity – curiosity exercised to no purpose. Though it seems that a sekha is not necessarily freed from this, for there was no evident benefit in Moggallāna being curious as to which devas knew of his attainment.
I agree with this.
A few thoughts and another angle from which to look at the OP.
I’d ask - What is the purpose of the curiosity? Curiosity is basically a desire and the question would be - what is being desired or craved? Knowledge and understanding about something… Does the thing craved lead to progress on the Path, or is it something that needs to be restrained. ie is it a desire about something necessary for progress or one not leading to the goal?
SO the pertinent aspect would be the object of the craving. To me it is this aspect that enables discernment of whether it is part of Right Effort and Right Mindfulness, Energy, Investigation etc. I feel that it is impossible to practice these aspects absent some curiosity… one is looking and investigating to see how the Khandas work in order to penetrate Delusion. This requires curiosity about how the khandas work, how delusion is generated in order to transcend delusion.
It is our desire to understand the human condition, or any state of Being, trapped in Samsara (suffering 1st NT) the causes of its arising, the causes of its cessation and how to implement, activate and follow the N8fP in order to end rebirth, then that desire (curiosity) is what is needed. Without it one would not question our condition and escape or Liberation would not be possible.
So if the object is aligned with this aim, is wholesome and about progressing on the path then that is beneficial and part of the training, but if it is not about this then it would fit into the ‘idle/frivolous’ or unwholesome categories.
Curiosity about other things is curiosity about the contents of Samasara rather than the mechanics of how and why we remain trapped in Samsara…
Of course this is neither good nor bad, it is just a question of whether it constitutes the training as outlined in following the Path.
Desire for knowledge about the world is an interesting thing to analyse, and I’d suggest this is directly linked to the Asavas (all 4). In the ultimate sense, if one is aiming for Nibbana, then all the Asavas need to be abandoned.
I’ll just mention that curiosity in order to develop and possess knowledge and views is something interesting to contemplate. Especially when considered together with Identity/Self view and Bhavasava, and could be seen as ponobhavika.
Note; in this reply I am talking about the goal of Nibbana, which requires relinquishment of this world. Not about other aims of improving or living successfully in the world.
So to posit an answer for the OP - What would an Arahant be curious about if the goal has been attained? Even when Enlightened, while one is interacting with the world, with Khandas intact, then functionality needs to be maintained. This functionality is linked to continued survival and beneficial wholesome interactions with others. Awareness, mindfulness and situational awareness must all be active… so whether you want to classify the force that drives where attention is placed as curiosity or not will determine how the answers are framed… I’m not convinced that this is such a useful framework to use in this circumstance.
Just something to consider 
That would be along the lines of tanhā vs. chanda, using the conditioned to progress on the path, wouldn’t it?
Another completely speculative idea comes to mind. Complete workings of kamma are probably not immediately totally clear even to an arahant, but they probably have a very good inkling that “something is up…” so to speak. If they foresee an event about to happen or for some other combination of reasons, I can totally believe that they might get curious to see how it plays out. I don’t have sutta references right now but there are plenty of stories of arahants and even the Buddha being at the right place at the right time for someone.