You will most likely have to remove the spider and check it for dry joints, thats a dash out job, if you remove your radio you will be able to see the unit from there, its about the same size as a pack of cards. They make it hard and time consuming to get to as its your car security and they didnt want anyone in the know to be able to just jump in and re mobilise it and drive it away quick as a flash.
if its the immobiliser spider would it be possible to input the EKA (emergency key access) code to disable the immobiser? drive around without the immobiliser off for a while and see what happens? I think it will remain disabled until you re arm it with the fob again. Someone more knowledgable will come along soon and tell you if this will work.
If it turns out it is the fault you can either sort out the spider (dash out) or buy a spider immobilser bypass unit (Landrover parts code: AMR 4956 for my model) apparently you can then just take the radio out unplug the spider leave it in place and simply plug in the bypass unit in about 10 minutes. Think it cost me about a tenner, havent got round to fitting mine yet as the problem is intermittent, I keep it in the glove box for peace of mind but if you ask me the job does look a bit fiddly (know anyone with little arms?)
Makes sense that it could be a "dry joint" in the spider, it gets hot and expands and sperates, it cools down and makes the connection again.
This is the fella who I bought my immobiliser spider bypass unit from, he was really helpfull and full of usefull information:
eBay My World - squikthecat
And here are previous posts of mine for you to look through if you like:
http://www.landyzone.co.uk/lz/f8/disco1-v8-immobilisor-problem-111605.html
http://www.landyzone.co.uk/lz/f8/immobiliser-bypass-question-111633.html
http://www.landyzone.co.uk/lz/f8/immobiliser-fun-167182.html
http://www.landyzone.co.uk/lz/f8/2-strange-things-my-keyfob-174897.html
They may not help you in the slightest but I thought they may be worth a look in the hope that they may give you a greater understanding of the Discovery immobiliser/spider issue