No, I don't think that's a fair distribution of tasks. It would be barely fair if both of you were working full time. If this is the split when he is at home all day I dread to think what it's like when he's working.
If you had children and he was doing childcare in the day then I'd say differently. But as it stands, that's not the case. Job hunting is soul destroying, but you have the luxury of fitting it in with what else needs doing. There is PLENTY of time to do small chores every day to reflect the fact you can't contribute in other areas right now. Do you guys live in an enormous house? That does make things take much longer.
We are in the same situation as you (reversed), and I don't think genders are relevant. It's about the fact that one person is at home all day (without children), and one person is at work all day.
My partner works pretty long hours, and tends to get home between 2000 and 2100 each day. On the chores:
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I do all the laundry. If partner is at home, he will often go and pick it up from the dryers, or help me carry it over (we don't have a dryer in the house). This is offered by him, not me asking. If I do it during the day then I do it all myself. I hate laundry and the dryers are expensive, so I tend to do it en masse once a fortnight. Partner will let me know if he is getting desperate for pants, but otherwise never "nags" me to do it in any way shape or form.
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I do all the cooking and washing up. Both of us set and clear the table together. Partner asks pretty much without fail if I want help with the washing up and I say no. He often offers to help with cooking, but I'm a better cook, and it's faster without him helping! He only gets very limited time at home and I prefer him to be able to relax.
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Partner always takes down the bins (glass, waste and recycling) and replaces the liner. They are heavy and he is stronger than me.
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I do all the cleaning in the flat, but admittedly our flat is very small. I do things as needed, to avoid having to do a really big job less often (like I do with the laundry). The flat is rarely spotless, but rarely a total tip. It's normally in a condition that I'd be happy (but not super proud) for people to visit and that's it.
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I do all the food shopping during the week because it's less busy and I like to plan what I'm cooking. Partner will go at the weekends if there's something we need or if we've run out of something unexpectedly.
Your partner seems to be doing the absolute bare minimum, and you are certainly doing more than him when you factor in that you work full time as well. If your partner wants to USE the bathroom then he has to be prepared to fucking clean it as well. He isn't royalty. I find it disgustingly precious that he isn't willing to "lower himself" by cleaning a bathroom he uses.
Likewise, why on earth are you sharing the hoovering? Why can't he run a hoover around during the day?
Look, your partner does need time to look for work, do applications, etc. But he can easily get up, do the washing up from last night, then make a coffee and sit at the computer. Then get up after a few hours and shove some clothes in the laundry. Then go for a run, or do the food shop, or whatever, and shove the clothes into the dryer (or hang them up) when he gets back. Then do more work for a bit. Then shove his earphones in and run a hoover around. Look, even doing everything I've set out above I still end up with more leisure time than I need.
I don't believe for a second that he doesn't take loads of breaks during the day do look at stuff on youtube, or facebook, or whatever else.
PS, Before anyone makes any stepford wife comments - As I said, I'm not working right now and we have no children yet. If the situation were reversed I would expect the same from my partner. If we were both working then no way in hell would I do this much!