The first thing is don’t panic. Start by working out whether your bill could actually be that high - so what kVA supply do you have?
Eg ours is 12kVA (60A), we’re on tariff bleu so lets say 15cents per unit. We can use a max of 12 units per hour (approx) so 12x0.15 per hour = 1.8€ per hour, 302.4€ per week or 7864.4€ in 6 months.
So, if you are on 12kVA or 9kVA you could, just, have an electricity bill that high - if you have a 6kVA supply (and you did say it was a small house) it is indeed not possible to use 5000€ of electricity in 6 months so that might be a starting point for the conversation with EDF. If by 6 months you actually meant “a bit more than 6 months” then it might be useful to know that a 6kVA supply running flat out can burn through 5000€ in just over 7 1/2 months.
Next question is does your meter genuinely think your use is that high - this is easy to check if you have a modern meter with LCD screen and flashing amber LED. There are three things to look at
How fast is the little amber LED flashing? It does one flash for each Watt Hour, you’d need a continuous load of 7.6kW to get through that much 'leccy, for which the LED will be flashing twice per second, all the time. You can also look at the number of units which will be going up by about 7 units per hour or (by pushing the buttons next to the display) the current passing through the meter which will be about 33-35 Amps.
It’s possible the load is not totally constant - especially if you do have a higher kVA supply as it could max out for part of the day then drop back (a 12kVA supply would need to be maxed out for 15 hours a day even if usage were zero at other times).
If the meter is flashing madly all the time then the next thing is to check is whether it is accurate. This might need the help of an electrician. I’d try to get a clamp meter around the live lead either going into or immediately out of the meter and check that the current matches the reading on the meter.
Assuming the meter is accurate the next question is what is pulling the power - this might need some further detective work with the clamp meter and tracing circuits but I’d start with a list of electrical items, especially heating, ballon’s etc which might be sufficient on their own, if faulty, to explain the high use. You mentioned a pool - is it heated? If so how.
If you can’t get an electrician out (or want to try to fault find a bit yourself) start by going to your tableau de distribution and turning every switch off - there should be at least one main RCD and then a smaller breaker per circuit. The amber light should stop flashing and the current should drop to zero amps. If not you need to figure out if anything else is wired into the meter. Then switch the RCD’s on. The current should stay at zero, finally go through the circuits one at a time switching them on and noting the current (Amps) reading. That might show which circuit is drawing a lot of power. Remember that something like a water heater might want to switch on if the water has cooled a bit while you had the power off (if you are on a split tariff normally the water heater should only be on overnight so if pulling the lot of power durng the day that might be a problem).
As for independent body - a good local electrician should be able to check things out for you.