I would go and ask my manager:
I am interested in Task F
but it has low priority at the moment, can I commit one hour daily to work on it?
If he says no because of the low priority, I would follow up with:
Can I work on it in my own time?
Lunch, weekend, after working hours?
This way you are transparent, expressing interests and/but "following orders".
See what he says and good luck!
workplace.stackexchange.com/users/51224/Sandra K
It is not unethical (to most people).
The more important question is would it be appreciated.
That would depend on your boss.
Some would think hey I got F for free.
Others may think not so sure A did not suffer here.
I would explore F and be sure you can do it in the free time you allot yourself.
If you are pretty sure you can get it done then ask your boss if you can work on it outside of work hours.
Tell him F is of interest to you.
workplace.stackexchange.com/users/26028/paparazzo
If you have to ask the question trying to get around the rule, you already know what the answer is.
However, it is not unethical to make that proposal to do the work extra hours on the task.
If you want to do work out of order you should get managements approval first.
If it really is just a matter of priority then getting that permission should be as simple as asking.
It is also possible that it was put in the priority queue where it was because their are decisions to be made about the task, and putting the task at the bottom of the pile was easier than making those decisions.
In that case your manager may not want you working on the project.
workplace.stackexchange.com/users/16/IDrinkandIKnowThings
A boss should be pleased that there is any aspect of your job that you would want to do in your free time.
Only a fool would discourage you from doing it.
Obviously, he may give the other tasks more priority and want those completed first, but when people do things they like, they feel better.
It should only help this project in the long-term.
An exception could be if this task is so far down the line in priority that it may get removed from the project.
Your boss should let you know that.
If it is going to upset you that this could be removed, he may suggest you don't do it.
Even though you're doing it on free time, it will still need to possibly be integrated in to the project and continuously supported in the future.
There may not be enough resources to maintain it.
workplace.stackexchange.com/users/None/
I'd say it's not strictly unethical, but as you say it might give a bad impression to your boss, then why risk?
Also, doing company work on your free time is not necessarily advisable, as it may harm productivity during work hours.
A good thing to do would be to discuss your proposition with the boss, and see what feedback you get.
You can also put it as a form of exercise, that you will have to manage yourself and possibly bill for it (as overtime hours).
workplace.stackexchange.com/users/15991/clabacchio
Take this as an opportunity to show initiative and interest, instead of being afraid of upsetting your boss.
Explain to him that you'd like to work on F (in your free time, if that's acceptable in the culture of your employer - otherwise, during a small percentage of the normal work day).
Make it clear that you already understand and follow the priority he has set, but that you think you can add value by solving F without impacting his priorities and F is a subject that you're already knowledgeable about.
It's fine to mention that you're personally interested, but I would also make sure there's a clear value-add for the employer, in order to give him incentive beyond just letting you work on a pet project of yours.
workplace.stackexchange.com/users/83999/dwizum
I am currently working on Task A, which is currently estimated to end in 2 months.There are five other tasks in pipeline: B, C, D, E and F.   Task F is something I care very much about, because it is related to my previous work in the company. However, it is the least important one to my boss. I am planning to work after office hours and in weekend (unpaid) to complete task F.  How can I properly approach my boss about working on a low priority task?
How can I approach my boss about working on a low priority task?
workplace.stackexchange.com
<professionalism><communication><ethics><work-time>
workplace.stackexchange.com/questions/109537
workplace.stackexchange.com/users/13113/user3070069
Discuss with your boss about working on F in your free time and listen to their feedback. Make it clear that you understand the priorities they have set but that you think you can add value by completing F without impacting them.  The answer to this question will really depend on the relationship you have with your boss, but you may wish to simply explain how strongly you feel about the task. Make sure they see that you understand the set priorities and make it clear that you're intending to work out of hours on task F so that it doesn't affect your other work.
The answer to this question will really depend on your boss and the relationship you have with them. When discussing this with your boss, simply explain that you understand the importance of the set priorities and that you are willing work outside of work hours on a low-priority task. Explain to him that you feel strongly about the task and can add value.