Mechanics has been a central focus of physical biology in the past decade. In comparison, how cells manage their size is less understood. In this talk, I will present a recent work in which it is shown that a parameter central to both the physics and the physiology of the cell, its volume, depends on mechano-osmotic coupling. Cells change their volume depending on the rate at which they change shape when they spontaneously spread, or when they are externally deformed. Cells undergo slow deformation at constant volume, while fast deformation leads to volume loss. This phenomenon is explained by a mechanosensitive pump and leak model. The model and experiments suggest that volume modulation depends on the state of the actin cortex and the coupling of ion fluxes to membrane tension. This mechano-osmotic coupling defines a membrane tension homeostasis module constantly at work in cells, causing volume fluctuations associated with fast cell shape changes, with potential consequences on cellular physiology.