The Rebellion Gauge is a gray meter with a black frame, which is present next to Joker's portrait at all times. By default, it has a red splash around it, but if the match is in Mementos and Persona 3 or Persona 4 music is playing, the splash's color changes to blue or yellow, respectively. As the Rebellion Gauge fills, the gray is replaced with pink.
The Rebellion Gauge slowly fills over time, but can be more efficiently filled by (in increasing order of effectiveness): one of Joker's teammates being KO'd, one of Joker's teammates being attacked, Joker himself being attacked (which causes a spark matching the color of the splash at the right end of the Gauge), and Joker being attacked while using Rebel's Guard.
When the meter is filled, Arsene is summoned and the Rebellion Gauge turns cyan with blue flames behind it. Arsene remains active to boost Joker's abilities until the Rebellion Gauge runs out, which takes 30 seconds. Taking damage causes larger portions to be removed. Once the Rebellion Gauge depletes fully, Arsene leaves Joker and the meter resets to its default state. Joker is invincible during the time Arsene arrives and leaves the field.
Several instances can temporarily halt the depletion of the activated Rebellion Gauge. These include:
The Rebellion Gauge fills in units, calculated by 1.3 * the damage Joker receives throughout the battle if he is hit. For example, a move dealing 3% will fill the gauge by 3.9 units. Taking around 76.92% (the actual calculation for this, 100 divided by 1.3, results in a very long number) will give enough units for Arsene every time. It will also passively generate units, at 0.00925 per Frame. In a more readable format, it fills at 1.11 units every 2 seconds; after 3 minutes spent without taking any damage, Joker will automatically summon Arsene. If a teammate takes damage, the gauge will also fill, calculated by 1.3 * the damage received by the teammate. If Joker uses Rebel's Guard, the gauge will fill by 1.3 * 0.4 * 6.7368 * the damage taken, instantly filling it after taking 28.6% damage or more. 
When the Rebellion Gauge reaches 100 Units, Arsene will appear for 30 seconds, or 1800 frames. If Joker is hit during this time period, the Rebellion Gauge will decrease by 12 * the damage received, provided two or three players are present, only this time in frames rather than units (to calculate in seconds, divide the result by 60). A move dealing 20% in an 1v1 battle will subtract 240 frames, or 4 seconds, from the Rebellion Gauge. Depending from the number of players in a match, the multiplier for the depletion of the Rebellion Gauge changes; it will use a 10.2 multiplier for a 4 players match, 8.6 for 5 players, 7.8 for 6 players, 7 for 7 players and 6.3 for 8 players. 
If Joker is KO'd at any point, the meter will reset at 20 units regardless of how filled it was beforehand. If Joker is behind in stocks, the gauge will fill at a faster rate. However, the specifics for this are still unknown. 
The Rebellion Gauge is functionally unique to Super Smash Bros.; in the Persona series, Personas could be summoned at will after their user has awakened them so long as the user has enough SP to spend for the selected attack, said SP functioning as it usually does in role-playing games. It has been theorized that the mechanic of the Rebellion Gauge is instead meant to invoke the feeling of awakening one's Persona, which in Persona 5 came from finding resolve in the face of extreme hardship. Thus, being on the defensive through most of the match could be seen as evocative of said moment, while Arsene appearing to grant Joker the power needed to turn the tide in his favor is evocative of him gaining the resolve to triumph over it.
When Arsene is active, the Rebellion Gauge's color fittingly matches that of the HP gauge in Persona 5. Otherwise, it matches the color of the SP gauge.