The favored progenitor model for short γ-ray bursts (SGRBs) is the merger of two neutron stars that triggers an explosion with a burst of collimated γ-rays. Following the initial prompt emission, some SGRBs exhibit a plateau phase in their X-ray light curves that indicates additional energy injection from a central engine, believed to be a rapidly rotating, highly magnetized neutron star. The collapse of this "protomagnetar" to a black hole is likely to be responsible for a steep decay in X-ray flux observed at the end of the plateau. In this paper, we show that these observations can be used to effectively constrain the equation of state of dense matter. In particular, we show that the known distribution of masses in binary neutron star systems, together with fits to the X-ray light curves, provides constraints that exclude the softest and stiffest plausible equations of state. We further illustrate how a future gravitational wave observation with Advanced LIGO/Virgo can place tight constraints on the equation of state, by adding into the picture a measurement of the chirp mass of the SGRB progenitor. Recent observations of long and short γ-ray bursts (SGRBs) show plateau phases in the X-ray light curves that last hundreds of seconds [1-6] and provide evidence for ongoing energy injection through a central engine [2,7,8]. The main candidate for the central engine in SGRBs is a rapidly rotating, highly magnetized neutron star (NS) [9][10][11][12] that forms following the coalescence of two NSs [13][14][15][16][17][18][19]. Recent analytic fits to X-ray light curves support this "protomagnetar" interpretation of a central engine for both long [20][21][22][23] and short GRBs [4][5][6]. Excitingly, some objects exhibit an abrupt cutoff in the X-ray flux ∼ 100 s after the initial trigger [5,20,21]. This has been interpreted as the metastable protomagnetar collapsing to form a black hole.From a theoretical perspective, the coalescence of binary NSs can follow a number of evolutionary paths. If the merger remnant is sufficiently massive, it immediately collapses to a black hole, or forms a dynamically unstable hypermassive NS that is supported by strong differential rotation and thermal pressure [18,24]. Magnetic braking terminates differential rotation on the Alfvén timescale [25,26] implying that the object collapses in ∼ 10-100 ms. If the merger remnant is less massive it forms a supramassive, metastable protomagnetar [27,28] in which centrifugal forces from uniform rotation support a higher mass than the nonrotating TolmanOppenheimer-Volkoff (TOV) maximum mass [29]. Such a supramassive star spins down until the centrifugal force is insufficient to support the mass, at which point it collapses to a black hole. The recent discovery of ∼ 2 M ⊙ NSs [30,31] demonstrates that the equation of state (EOS) permits massive enough NSs for supramassive stars to be created from the merger of two NSs [32]. Finally, a merger remnant that is less massive than the TOV maximum mass will survive as a stable NS.In this paper, we focu...