So this is the second time I've had to add a whole entire jug of coolant to my reservoir tank and I don't see any leaks and the guy at the mechanic Place told me that this is normal that coolant evaporates because it gets hot can someone tell me if I should bring it in because I also smelled burned coolant

 

There is nothing normal about having to add 2 full gallons of coolant, and it's a closed system so there really should be no evaporation unless there is, in fact, a leak.

If you're smelling it but can't see it (assuming there is no coolant on the ground either), it could be the water pump or head gasket. It could also be one of the many plastic tubes/pipes on top of the engine underneath the supercharger, and the coolant is pooling/burning off without leaking to the ground.

I'd definitely take it into shop and have it pressure tested for starters.

 

I saw an advert for an XFR for sale in Canada last Summer and the seller stated; "all coolant lines replaced because of known problem with XFR, although no leaks."

There must be more to this story that someone knows.

 

The XF has an undertray that catches most leaks before they hit the ground (at least my 2012 does). You need to find a different mechanic. We know the 5.0L engines have some faulty design issues around their water pumps with Jaguar finally just giving up after version #8 and now aftermarket pumps available from Beck Arnley and others that seem to hold up a little better. But there's also a lot of PA66 nylon components used in the Jaguar cooling system, a material that never should be used in cooling system since it actually dissolves in ethylene glycol at very high temperatures. At normal operating temperature its not supposed to, but over time and little temperature spikes here and there eat away at it little by little. At this age, ALL of the PA66 nylon components in the cooling system should be replaced, and again, the aftermarket produces some of them in cast aluminum. A good independent Jaguar mechanic will know all of this.

Of course there are lots of hoses, clamps, o-rings, gaskets, other seals and other components that need to be checked too. What they'll do is hook up a device and pressurize the cooling system so that they can look for active drips. That way they can get up real close to safely inspect for leaks without burning themselves or getting their gloves caught in the moving parts and they can take things apart/move them out of the way so they can see. If its leaking gallons, then its going to leak whether its hot or cold and it'll be obvious once they do the pressure test.

 

My XJ (X351) started losing coolant when the radiator fan stopped working in the summer. Engine temps were pretty high (up to 225/230F) at stop lights, but the car wouldn’t tell me. Coolant started boiling off. It was the repeated need to fill the coolant despite no leaks, good pressure test, and no exhaust fumes in the coolant that made me check engine temps. Replaced fan and no issue anymore.

 

Pressure test coolant system when cold.
You have a GIANT leak somewhere so it should just pour out.

But do listen to Rob19321! ALL of your hoses are shot at this point and you will be MILES ahead to just bite the bullet and start gathering parts. Along with a new water pump unless you have replaced yours already?
The car is over 12+ years old so if you plan to keep it there will be maintenance costs. With a 2010 I would be checking the timing gear stuff as that will take you engine out pretty quick.

If you can give us ALL the car info it makes it much easier. You were great but miles matter too?
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