Wouldn't the thermodynamic cost of creating alternate universes make the Many Worlds interpretation implausible?

The many worlds interpretation has a lot of problems, but this isn't one of them. You're imagining "creating alternate universes" as some energetic event, like a mini Big Bang, but what really happens is a smooth splitting of the wavefunction.

For example, suppose we have a spin in a superposition of up and down states, $$\frac{1}{\sqrt{2}} (| \uparrow \rangle + | \downarrow \rangle).$$ Now, suppose we measure the spin with a macroscopic detector that displays $1$ if the spin is up and $0$ if the spin is down. Then under the Copenhagen interpretation, the final state is $$|1, \uparrow \rangle \text{ or } |0, \downarrow \rangle$$ each with 50% probability. Under the many worlds interpretation, the final state is $$\frac{1}{\sqrt{2}} (|1, \uparrow \rangle + |0, \downarrow \rangle)$$ and these two terms are the two "worlds". Energy is still conserved. Though there are now two branches of the wavefunction, each has $1/\sqrt{2}$ times the amplitude, so half the energy.

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All standard interpretations of QM agree this is the right final state for two microscopic objects interacting; many worlds just extends it a macroscopic object. The difference is that interaction with a macroscopic system is irreversible by the Second Law, so you'll never be able to unentangle the detector and get back to your original state.

As a result, we call the two branches of the wave function different "worlds", since they can never interfere with each other. More conservative interpretations, like Copenhagen, just say that the other world doesn't exist at all. It's a matter of taste, since the difference is unobservable.