Be it humans or machines under extreme conditions, both struggles, with the latter more vulnerable to failure. Both are designed to function/sustain within a specific temperature range. So is the case with the energy sources- Battery. Take, for instance, a automobile grade battery; it is especially susceptible when operating in adverse conditions. This is why these batteries dies on a chilly winter morning, albeit it worked fine the previous afternoon.
Fig 1: effects of extreme temperature on car batteries 
The standard rating for batteries is at the temperature of 25 degrees C (about 77 F). At approximately -22 degrees F (-30 C), battery Ah capacity drops to 50%. At freezing, capacity is reduced by 20%. Capacity is increased at higher temperatures at 122 degrees F; battery capacity would be about 12% higher.
A car battery operates best when the air temperature is 80 degrees Fahrenheit (26.67 °C). It is said that the temperature under the hood will increase once the car’s driven several miles and stay at an elevated temperature for the trip’s duration. That is why when a heatwave arrives (90 degrees Fahrenheit (32.22 °C) or more), you will see more vehicles under threat.
At higher temperatures, the battery's cyclic life would significantly reduce. If, for example, A battery operates at 30 °C (86 °F) rather than a more moderate lower temperature, the cycle life is reduced by 20 percent. At 40 °C (104 °F), the loss jumps to a whopping 40 percent, and if charged and discharged at 45 °C (113 °F), the cycle life is merely 50 % of what is often expected if used at 20 °C (68 °F).
An increment in the environmental temperatures will corrode the battery's internal components leading to performance degradation and even weaken its power and capacity. The heat might not cause outright failure, but it will set the battery up for failure later. With the lead-acid battery, there's the danger of the electrolyte freezing, which may crack the enclosure. Lead-acid freezes quicker with a coffee charge when the precise gravity is more like water than when fully charged.
High-temperature conditions accelerate the thermal aging and may shorten the lifetime of batteries. Most of the time, high-temperature effects are attributed to batteries' high internal temperature during operation rather than the environmental temperature. The high internal temperature is caused by heat generation inside the batteries, which happens at a high current state, including operations with a fast-charging rate and fast discharging rate.
Countries like Russia, Canada, and Greenland Island are generally high-altitude places with shallow temperatures throughout the year. Batteries will show slow chemical-reaction activity and charge-transfer velocity at these low operating temperatures, which leads to the decrease of ionic conductivity in the electrolytes and electrode-ion diffusivity. Such a decrease will result in the reduction of energy and power capability and sometimes even performance failure. At –20 °C (–4 °F), most batteries are at about 50 percent performance level. Although the NiCd battery can go all the way down to –40 °C (–40 °F), the permissible discharge is merely 0.2C (5-hour rate). Specialty Li-ion can operate to a temperature of –40 °C but only at a reduced discharge rate; charging at this temperature is out of the question.
Specifically, a battery operating at 100% capacity under optimum conditions will lose about half its strength when the temperature reaches 0 degrees Fahrenheit (-17.78 °C). For older and weaker batteries, capacity is already reduced, which means a cold snap can do them. Increased pressure on battery life also occurs when temperatures retreat and forth between optimum and extreme conditions.
Behavior of Battery with temperature variances:
To control the operating temperature of batteries and ensure the performance and safety, various battery thermal management systems (BTMSs) are designed for thermal management. However, it is difficult to monitor the temperature distribution inside the batteries, which are tightly sealed during operation.
The current cooling strategies are mainly based on air cooling, liquid cooling, and phase change material (PCM) cooling for the batteries working under high-temperature conditions. Liquid cooling seems to be more effective and attracts researchers mainly due to its higher heat transfer coefficient.
Depth of Discharge:
A battery's depth of discharge (DoD) is the percentage of the battery used up or discharged concerning the battery's total capacity. This DoD has a direct relation with the battery's lifespan and even the number of discharging/ charging cycles it can undergo or, in a single word, the 'cycle life.' The more frequently a battery is charged and discharged, the shorter is its lifecycle. It is generally not recommended discharging a battery entirely, as that intensely shortens the battery's usable life.
Manufacturers, at times, mention a maximum recommended DoD for optimum performance of the battery. That is, if a battery of say 12KW-h has DoD as say 68 %, then about 8.160 KW-h is the recommended discharge that the battery can undergo before its recharge. Any discharge below this value will eventually lead to a decrease in the cycle life of the battery.
Fig 3: the above figure shows plot of battery capacity versus number of cycles at varied DoD% 
It can be observed from the above plot that, when several batteries are considered with different Depth of Discharge %, the one with the highest percentage that promises a longer one -time charge period has the least cycle life , and it becomes exactly the opposite when we are ready to give up on the DoD %; we are returned with highest possible cycle life for that capacity. That means DoD% and cycle life of battery come at the cost of each other.