  We study a Cooper pair transistor realized by a mesoscopic superconductor
island that couples to a pair of $s$-wave superconducting leads. For a trivial
island, the critical supercurrent between the leads exhibits a well-known
$2e$-periodicity in the island-gate charge. Here, we show that for an island
with spatially separated zero-energy Majorana or Andreev bound states the
periodicity of the magnitude of the critical supercurrent transitions to $1e$
in the island-gate charge. Moreover, for Andreev bound states the current-phase
relation displays a sign reversal when the parity of the charge ground state of
the island changes between even and odd. Notably, for Majorana bound states the
same sign reversal does not occur. Our results highlight the relevance of
measuring the full current-phase relation of a Cooper pair transistor for
clarifying the nature of zero-energy bound states in candidate systems for
topological superconductors and provide an initial step towards integrating
Majorana qubits in superconducting circuits.
