Feedbacks that can destabilize the climates of synchronously rotating rocky planets may arise on planets with strong day-night surface temperature contrasts. Earth-like habitable planets maintain stable surface liquid water over geologic time. This requires equilibrium between the temperature-dependent rate of greenhouse-gas consumption by weathering, and greenhouse-gas resupply by other processes. Detected small-radius exoplanets, and anticipated M-dwarf habitable-zone rocky planets, are expected to be in synchronous rotation (tidally locked). In this paper, we investigate two hypothetical feedbacks that can destabilize climate on planets in synchronous rotation. (1) If small changes in pressure alter the temperature distribution across a planet's surface such that the weathering rate goes up when the pressure goes down, a runaway positive feedback occurs involving increasing weathering rate near the substellar point, decreasing pressure, and increasing substellar surface temperature. We call this feedback enhanced substellar weathering instability (ESWI). (2) When decreases in pressure increase the fraction of surface area above the melting point (through reduced advective cooling of the substellar point), and the corresponding increase in volume of liquid causes net dissolution of the atmosphere, a further decrease in pressure will occur. This substellar dissolution feedback can also cause a runaway climate shift. We use an idealized energy balance model to map out the conditions under which these instabilities may occur. In this simplified model, the weathering runaway can shrink the habitable zone and cause geologically rapid 10 3 -fold atmospheric pressure shifts within the habitable zone. Mars may have undergone a weathering runaway in the past. Substellar dissolution is usually a negative feedback or weak positive feedback on changes in atmospheric pressure. It can only cause runaway changes for small, deep oceans and highly soluble atmospheric gases.Both instabilities are suppressed if the atmosphere has a high radiative efficiency. Our results are most relevant for atmospheres that are thin, have low greenhouse-gas radiative efficiency, and have a principal greenhouse gas that is also the main constituent of the atmosphere. ESWI also requires land near the substellar point, and tectonic resurfacing (volcanism, mountain-building) is needed for large jumps in pressure. These results identify a new pathway by which habitable-zone planets can undergo rapid climate shifts and become uninhabitable.