A sleeve (10; 30; 60) for a beverage cup, the sleeve comprising: a main wall (12; 32; 62) having an open top end and a bottom end, the main wall having an inverted truncated conical shape and defining a central cavity for receiving at least a portion of a beverage cup inserted into the central cavity via the open top end; and a bottom (14; 34; 64) hingedly attached to the bottom edge of the main wall (12; 32; 62), the bottom comprising a circular floor portion and a plurality of fingers (18; 38; 68) projecting radially from the floor portion; the sleeve (10; 30; 60) being movable between a flattened state for storage and transport in which the main wall (12; 32; 62) is flattened such that opposing sides of the main wall are in contact with each other and the bottom (14; 34; 64) is folded toward the main wall and positioned between the opposing sides of the main wall, and a deployed position for use in which the bottom is substantially perpendicular to a longitudinal axis of the sleeve and closes off the bottom end of the main wall (12; 32; 62), characterized in each of said fingers (18; 38; 68) having a trapezoidal or triangular shape and being coplanar with the floor portion when the sleeve (10; 30; 60) is in the flattened state, and, when the sleeve (10; 30; 60) is deployed from the flattened state into the deployed position, the bottom (14; 34; 64) moves away from the main wall (12; 32; 62) and downward toward the bottom end, and the fingers (18: 38; 68) fold upward relative to the floor portion and are in contact with an inner surface of the main wall (12; 32; 62) without the fingers interfering with each other when the sleeve (10; 30; 60) is in its deployed position for use. a main wall (12; 32; 62) having an open top end and a bottom end, the main wall having an inverted truncated conical shape and defining a central cavity for receiving at least a portion of a beverage cup inserted into the central cavity via the open top end; and a bottom (14; 34; 64) hingedly attached to the bottom edge of the main wall (12; 32; 62), the bottom comprising a circular floor portion and a plurality of fingers (18; 38; 68) projecting radially from the floor portion; the sleeve (10; 30; 60) being movable between a flattened state for storage and transport in which the main wall (12; 32; 62) is flattened such that opposing sides of the main wall are in contact with each other and the bottom (14; 34; 64) is folded toward the main wall and positioned between the opposing sides of the main wall, and a deployed position for use in which the bottom is substantially perpendicular to a longitudinal axis of the sleeve and closes off the bottom end of the main wall (12; 32; 62), characterized in each of said fingers (18; 38; 68) having a trapezoidal or triangular shape and being coplanar with the floor portion when the sleeve (10; 30; 60) is in the flattened state, and, when the sleeve (10; 30; 60) is deployed from the flattened state into the deployed position, the bottom (14; 34; 64) moves away from the main wall (12; 32; 62) and downward toward the bottom end, and the fingers (18: 38; 68) fold upward relative to the floor portion and are in contact with an inner surface of the main wall (12; 32; 62) without the fingers interfering with each other when the sleeve (10; 30; 60) is in its deployed position for use.