Do all vacua in the string theory landscape have a different cosmological constant?

Question

Or can two vacua with the same energy differ in other ways?

Answer 1

The string one-loop cosmological constant is given by $$\Lambda=\int_{\mathcal{F}}\frac{d^2\tau}{(Im \tau)^2}Z(\tau)$$ where the integration is over the Teichmuller space of inequivalent tori and $Z(\tau)$ is the partition function of the theory. The partition function keeps track of how many particles with any given mass appear in the spectrum of the theory, but not of their interactions. So, it can happen that two different vacua have the same partition function and therefore give the same cosmological constant. This was also discussed in the post "Same partition functions, different theories".

NB: That being said, note that supersymmetric theories formally have $Z(\tau)=0$ since the contributions of bosons and fermions exactly cancel each other, so if you are only interested in such vacua I see nothing wrong with treating them as if all give the same cosmological constant (zero).