What is the best response when a husband with dementia threatens divorce?

Then follow it with "Oh, you said divorce, I thought you said intercourse"

“All right, when I get a minute, I’ll get you a list of divorce attorneys in the area. Now, do you want a ham or turkey sandwich for lunch?”

"Not tonight honey, I have a headache. "

Say....
Then lets sit down and talk about this.
Get out a pen and paper and sit down together and start going through all the things you have done and use this opportunity to "divide" memories.
You can start by...I remember when we met...and write down your thoughts about that. He may respond by saying I remember...my first car, first job....and write down his thoughts.
This is also a good way to possibly pull memories that you were not even aware of.

Next time he mentions this you can "divide" the items in the room..get him to name things in the room he "wants".

Some of this you can say...Ok, lets talk about it after lunch, or dinner...
You can laugh about it.
Sometimes you can say...oh, let me think about this and leave the room for a moment.
There were times when my Husband was doing things I would try to distract him from doing and once I just broke out into a laugh..a real belly laugh and for some reason that stopped him in his tracks and he just started laughing as well. And funny thing was that was the first time I had heard him really laugh in quite a while. His mood changed for quite a while. So I would sometimes just laugh for no reason and it did the same thing. (maybe laughter is the best medicine)

But please, I hope this does not hurt your feelings or that you take this seriously even a little. Yes it might hurt for a second but I am sure you know deep down he does not mean what he is saying.

Is he capable of being left alone for short periods? If he still has the capacity to know what he is saying I would tell him (gently) that you are not going to argue and leaving for a bit to let him calm down.
You could go for a ride around the block, take a short walk, or even just go outside for awhile.

It's impossible to stay mad for long at someone who doggedly remains loving and considerate. Respond with 'I'm sorry' (you feel that way). Then get out of his way.

By the way on a very serious note here.
Some types of dementia can cause a person to become violent. If this ever happens..even just 1 time you need to consider placement for him.
A very dear friend of mine was seriously hurt by her husband and to this day has physical problems. The proverbial straw that broke the camels back was when she woke up when he had his hand around her neck and a knife in the other hand. She realized then that this could not continue but it was a while before she could get her children to accept the fact that their dad was doing these things.
I always said I would keep my Husband at home as long as it was safe for him and safe for me. This meant any kind of safe. Physical safety was just 1 aspect.

Ok, off my soap box for now.
Just be safe, think about who will care for him if you are injured or worse...and who will care for you..

I too have the same issues. My hubby of 29 years has a long history of failed relationships and "man is in charge" attitude. He has progressing dementia caused by 10+ years living with PD. The cognitive decline is now going on 3 years. I have had to become the only adult in charge of our lives, finances, and futures. I am is main caretaker. He frequently accuses me of stepping out when I go to the grocery store or have a 3 hour respite movie and coffee with a girlfriend. He has asked for a divorce thousands of times. I just redirect and wait for it to blow over. I say, "ok, I'll get an attorney and we will work on splitting everything. This will take about a month or two. Your son will need to step up and take you in and be responsible for your care and finances. Let's have some ice cream right now." Then he ponders on that for a few hours, takes a nap, and forgets. Other days he tells me that he is glad he is married to me. Other days he thinks I'm just another caregiver. Some days he thinks I'm against him and some days, for him. Just part of the brain not working properly. I'm thankful he is not violent, just paranoid. I have a support group, see a therapist, and give myself a break by working and socializing. It is very difficult and exhausting, but we need to take care of ourselves or we can't take care of our loved one.

Just diffuse it with something like..."why don't we discuss that later tonight. Lets have a coffee (tea or whatever) right now". Or just leave the room and close the door so he can't get to you. Let him spew and then hopefully settle down.

Once I read a response from a wife when her husband with ALz threatened to kill her said "ok, but let's have lunch first"...then he forgot.
I'm so sorry - it must be sad to have him say this to you.

Lots of good answers here. There is one that hasn't been suggested, though, and in my opinion, well worth a try. I don't know what your faith system is, but have you considered praying about it, both on your own and with him? You might respond by saying, "Honey, do you really think we should get divorced? I love you and believe that you love me. I don't want a divorce. Let's pray about it." Then if he will le you, take his hands in yours and pray something like this: "Heavenly Father, thank You for giving me this wonderful man as my husband and life partner. You have blessed our marriage all these years and I am asking You to bless us now. Please give us wisdom to know what to do and help us to honor You with our decision. In Jesus' name we ask. Amen." Keep it very short, simple and to the point.
I believe God will bless you.