There is no permission needed to go to temple by the policies or doctrine of the Church of Jesus Christ of Latter-day Saints. 
 
The only requirement needed is to hold a current temple recommend, which is obtained by a worthiness interviews conducted by the Ward Bishop and Stake President.  
 
What we do believe though is that the husband is the head of the household. That he is the leader of the family, in a household where the husband holds the priesthood he is the patriarch and priesthood leader of the family.  
 
The Husband is not a king though, husband and wife work together to provide to there family physically and spiritually.  
 
The most important organization in the LDS Church is the family. So the last thing any member wants to do is make choices to put stress or cause problems in the family unit.  
 
Your wife isn't required to ask your permission to attend the temple but I think she wants your approval to confirm that here religious practices will not create tension in your marriage. 
 
I cannot say what your wife's intentions are or why she asked permission to go to temple. I can tell you that our Church does not require a wife to ask permission of a husband, it does encourage that husband and wife work together and not do things to create animosity or problems in the marriage.  
 
Maybe this is why she sought your approval but I cannot say for sure. Do know though that as far as I know there is no husband permission needed to attend temple.  
 
Thank you for the question I hope my answer helps!

I want to put a little different spin on this. 
 
As a priesthood leader (Bishop and Stake Presidency member) who will be the one to sign off on a temple recommend - I would be very reticent to sign a recommend for a sister unless I know that the non-member husband is ok with it. Does that mean that the husband has to give written approval? No. But I would as part of the interview with the wife ensure that she has thoroughly discussed with the the husband so that he knows exactly what to expect afterwards (clothing standards, tithing expectations for her, attendance expectations, etc.). In fact, if it were me I would prefer (but not require) to have a discussion with the husband about this myself. However, as I understand it - it is not a written down requirement. But I might be wrong.