To extend on the comment about heat-pipes and add in some answering-powder, real quick before I go back to work:
If you put a Peltier element on something, then connect a heat-piping mechanism to that, then connect another Peltier element at the other side of the heat pipe, you will be "pumping much faster".
Any system or material that transports heat does that somewhat reluctantly (some extreme and currently real-world-infeasible examples excluded).
This reluctance can be expressed as a thermal resistance. The flow of thermal power depends on the thermal resistance and on the thermal potential in that resistance, just like in electronics.
So to make the heat flow faster, you can decrease the resistance:
- Increase the flow of oil or water in a coolant system
- Make a metal heat-pipe thicker
- Blow on the object in the case of air-transfer.
But you can also keep the resistance the same and increase the potential, this "potantial" is expressed in a thermal difference. So if one end of the heat-pipe (use that for the example) is 50degrees C, and the other is 30degrees C, the differential is 20 degrees and the heat will flow with speed x.
If you now heat up one end of the heat-pipe to 70degrees C and cool down the other to 10degrees C, you have a differential of 60 degrees and if the resistance stays the same (real-world: not exactly, but, in a margin...) your thermal energy will be able to flow at 3x speed.
So, if you have a device that is 50 degrees C, and an environment of 30 degrees C, but need the heat to flow through a heat-pipe between them as quick as if there's a difference of 60 degrees C, then you can put a peltier on each end, capable of a thermal differential of 20degrees C under the circumstances.
Other than that, another answer would have been (as was already hinted at above):
Just use a pump, some hoses and a decent type of oil: Done! :-)
EDIT: Due to comment-y adjustment of the parameters, more options:
You asked/clarified in a comment: "Yes, but what if, temperature goes through a wire as electrons and comes out as temperature on the other side?", right?
Well, that is both possible and not possible. It certainly isn't feasible in any big way, but it is actually possible to test that in the real world. You will just be wasting a shit-load of energy until you Eureka yourself a better system that nobody's though of before with much higher efficiency. Though I'd be very surprised if you get actual mentionable numbers for the efficiency. But I'm happy to be surprised most days, so have at it.
What I'm getting at is the following fun fact:
A Peltier element doesn't just work one way. You can also use it to generate electricity from the thermal differential. If you apply boiling water to one side, and run room-temperature water along the other, that will induce a voltage. Its maximum current and voltage won't be super impressive, but they're there.
So, for the thought experiment, imagine you have the Peltier element neatly connected to the ground, which is mega-thermally conductive in your back-yard and its temperature is -20 degrees Celcius, because it's winter. Or you live somewhere else than you think you do, whichever works for you, just imagine it.
So, that side is always -20 degrees C, because the ground will take away anything you throw at it. Yet, your feet are warm. And also made of thermally super-conducting metal - I mean, I'm already claiming your backyard is a thermal-super-conductor at some fixed temperature, so I might as well just claim that as well. Oh and if I'm simplifying a bit further, your body generates heat infinitely fast if your foot-temperature drops a 1/100th of a degree.
((Small side note: Imagine having to accept all that ^^. .... That's how newly become professionals feel when people simplify op-amp or transistor characteristics. ;-) ))
Done imagining? Good.
Now, imaginarily step onto the Peltier. Hurray! Voltage!
You can step off now, since nothing is getting cooler or warmer, we can easily get our ever-ready spool of super conducting AWG30 wire from the garage, all the parameters will be just the same in a little while.
Now, obviously, while at the garage, don't forget to look into the never-ending box of useful stuff and get out another Peltier element. There's infinitely many in there anyway.
Hook up the second Peltier element to the first one, walk to your front yard, to prove that it works you always need an imaginary house in between set-ups. It's a well known fact.
Put down the element there, such that its cold side is on the ground, which there also happens to be thermal-super-conductor at exactly -20 degrees C.
Now the fun stuff really starts: Because this is a thought experiment we can also just say that a Peltier Element is always 100% efficient, regardless of how you use it and it will magically be so. Let's also say that the element can wick away a couple of kilowatts of energy at the temperature differential of
37 vs -20 = 57 degrees C.
Now we have made ludicrous assumptions -which we are allowed to- about everything in this experiment, except for the basic functionality of the inner-guts of the Peltier element. And that's what it's about.
So, now step onto the element in the back yard: Poof! The snow in the front yard just started melting! QED.
Oh, did I forget to tell you? While you were walking back it snowed a meter thick exclusively in your front yard.
Exiting the imaginary world that I nearly permanently host on the inside of my head
Of course my little magic world inside my head, and hopefully yours a little now too, doesn't mean that much in the way of a promise of functionality. But it does help imagine the theory behind it.
When you stepped on the plate in the backyard, electrons were carried along the junctions because of the forced flow of heat. Those electrons shuffled along our wire to the other one and carried energy out from the earth into the plate, radiating it into my layer of super-fast-falling snow.
Obviously, if you did this even with a large peltier element and both feet, you'd be lucky to get any humanly detectable differential at all on the other element.
So, inside my head: Yup, your idea works. For now, unfortunately only there. Or at least, feasibly.
For a fun look at some real-world stuff youtube "thermo-electric generator", it'll probably get you a lot of steam-powered stuff, but I remember a few years back seeing some videos of people using TECs (thermo-electric coolers / peltier's) to make a small fan spin with boiling water versus tap water in the winter and things like that.
At the time I was thinking about a tongue-in-cheek gizmo that charged a cap from thermo energy that then pulsed its power through a diac/triac into a 230V pilot valve closing the cold water tap, but I never actually did any maths to see how much fresh water I'd be wasting before the tap got turned off.