Temperature is one of many parameters that influence the rate of film development. Manufacturers standardize their adviced development times under the assumption you are developing at 20 C / 68 F, and they supply handy tables and graphs such as this one by Ilford (thanks for the link, Erik Gould!) to correct the development time if your temperatures are off.
On Twitter, Dev Samaddar posted the following the other day:
In this post, we will explore this topic further and try to answer both questions.
Is the temperature-time relation linear?
Let’s start with the second part of the question: “Is the temp-time relationship linear?”
A chemical reaction can only take place when the particles collide. The higher their kinetic energies, the higher their velocities and the more frequent collisions occur. However, if the collisions are not violent enough, nothing happens. In other words, we need frequent collisions of an energy that exceed the minimum amount of energy required to get a reaction going. This energy level is called the activation energy. The underlying physics dictates that the probability of a particle having a certain kinetic energy is described by a Maxwell-Boltzmann distribution. The higher the temperature, the higher the probability that a particle exceeds the required activation energy.
The relation between temperature and increased probability is exponential. In the past, a rule-of-thumb was observed that the reaction rate increases by a factor of two for each 10 C temperature rise. For this to be valid, the system is assumed to be homogeneous, at ordinary/room temperature, and the reactions should last longer than 2 seconds. All are approximately valid for the development of photographic materials. However, this rule-of-thumb turned out to be very approximate indeed, and factors between 1.5 and 4 were also observed for every 10 C . Others suggested that the rate increases by 5% – 10% for every 1 C in temperature rise; the equivalent of factors of 1.6 – 2.6 for every 10 C.
The reaction rate can be better approximated using the Arrhenius equation:
where factor and the activation energy are assumed to be constant, and is the absolute temperature. We see here that the rate is a function of temperature. However, in developer solutions many reactions are ongoing at the same time, and the slowest step determines the rate. These rates may change as development progresses, or the rate-determining step may change too. This makes it only possible to apply Arrhenius equation to single steps in the development process, and researchers have measured activation energies between 0.12 kJ/mol and 9.6 kJ/mol for various steps in the development process, where the values at the low end are dominated by diffusion and values at the high end are associated with fog development .
For developer solutions we can use it to discuss some more general observations. Mason  mentions that the following law holds for temperatures as low -25 C:
where is the time to obtain a given gamma, is the temperature coefficient, and is again the absolute temperature. The temperature coefficient depends on the developer composition and the temperature. It increases with higher bromide concentration, but decreases with pH. In general terms, Jacobson and Jacobson stress that in particular the quantity and alkali also play a part in the temperature coefficient.
Because many reactions are ongoing at the same time during development, and multiple developing agents can work together in what is called superadditivity, finding a temperature coefficient for a developing solution is difficult. As John and Field highlight, an alteration in development time cannot fully compensate for a change in temperature, especially when more than one developing agent is considered as their activities will change differently with temperature changes. For example, a standard MQ developer containing both metol and hydroquinone will see a shift from metol being more active at 10 C to hydroquinone being the more active of the two at 30 C . This will result in a change in contrast with temperature too, that is not easily compensated by a simple change in development time. If you need to work at temperatures that differ greatly from the specified temperature, you are better off with running a film test or using a developer that is designed for use at these temperatures, instead of just correcting the development time.
Why 68 F / 20 C as a reference?
I couldn’t find sources that cite a specific reason for this. However, in many fields of engineering and science 20 C is the agreed value for room temperature, although values between 15 C and 25 C are common too. As we will see below, it is also a temperature that many of the processes behave well in: the gelatin isn’t too soft; the developing rates are not too high or nor too low; most ingredients are soluble at 20 C; and it is a temperature that can be obtained in many home environments without too much trouble. Interestingly, some manufacturers of photographic products also state 24 C as a reference temperature. Likely because it is easier to heat up chemistry to a value above room temperature, than it is too cool it.
What does this all mean in practice?
Even though the relationship between temperature and time is not linear, correcting for small temperature changes is not as difficult as it may sound. For a range of roughly 10 C, the rules-of-thumb mentioned above work fine (that is, between 15 C and 25 C).
If you are in a pinch or only need a correction very seldomly, reduce the development time by 10% for every 1 C of temperature rise and you will likely reach a satisfactory result. If you want a better estimate, reduce the development time by a factor of 2.5 for every 10 C in temperature rise for Kodak D-23 and D-76 or Ilford ID-11 . You can use a factor 2.88 for most standard MQ developers .
For those less math-savvy, this means that you have to do the following operations. Assume you need a time at 20 C and need to find the developing time at a temperature of 22 C. In that case you just increase by 10% twice (not 20% at once, although the error is small for small temperature deviations).
If you live in an area where room temperature never reaches as high or as low as 20 C, then it is best to perform a film test and standardize your process on your average room temperature. Or as Akin puts it:
When in doubt, estimate. In an emergency, guess. But be sure to go back and clean up the mess when the real numbers come along. – Akin’s 10th law of Spacecraft Design
As mentioned earlier, this will also take into account the change in contrast and give you a considerably better estimate of the required development time.
But what if it’s very hot or very cold?
At low temperatures, the simple answer would be to just heat it up to 20 C. Most developing agents become ‘inactive’ below 12 C  (i.e., they become very slow, there is still some activity). If you need to develop below these temperatures and there is no way of heating up your chemistry, start with a developer that is already very active and use it undiluted. For example, an Amidol-Pyrocatechine developer can work down to -15 C if needed . Both ingredients are very active at room temperature, and can work together to make a developer that still works reasonably well at low temperatures.
At very warm temperatures, we run into other problems: developers become very active; there is an increased risk of excessive chemical fogging; and the gelatine may swell and become very weak  making the film very sensitive to damage. The gelatin may start the frill or to even disconnect from the support . In this case, you need to use a pre-hardener to harden the gelatin (can be used up to 32 C). Also, you may need to add an extra anti-foggant to the developer to inhibit chemical fogging.
A normal developer can be made suitable for tropical temperatures by adding a significant amount of sodium sulphate to slow down the developer. The sodium sulphate will also greatly inhibit the gelatin from swelling. According to Jacobson and Jacobson, you will need 135 g of sodium sulphate for every 1000 g of working solution of the normal developer solution. This is in line with what John and Field suggest for anhydrous sodium sulphate (10%) . When using the hydrated version, you need to add 23% in place of the 10%, because of the water that is included in the molecules. The included water will further dilute the developer, making it necessary to extend the development time by approximately 10% again to compensate. With this, you will be able to develop up to temperatures as high as 41 C and still have reasonable development times that are not too short for manual processing. If you want to adapt an exisiting developer, Anchell says that “a mildly alkaline, buffered borax developer is recommended. Alkali-free developers of the amidol type or one of the mildly alkaline fine grain developers such as Kodak D-23 are preferable to those with normal alkali content.” 
Steve Anchell  and Jacobson & Jacobson  also provide formulae for specific tropical developers in case you need them. I recommend you get their books and try out the formulae to see what works best for you. You will need to be patient though, as you will have to source and mix the individual ingredients yourself. The developing times for these tropical developers are still fairly short though and can be a mere 2 minutes at temperatures of 32 C.
 I.A. Leenson, “Old Rule of Thumb and the Arrhenius Equation”, Journal of Chemical Education, 76 (10), p. 1459 – 1460 (1999)
 R.W. Lambrecht and C. Woodhouse, Way Beyond Monochrome, 2nd edition, Burlington, UK: Focal Press, 2015.
 L.F.A. Mason, Photographic Processing Chemistry, London, UK: Focal Press, 1975.
 K.I. Jacobson an R.E. Jacobson, Developing: the negative technique, London, UK: Focal Press, 1976.
 S. Anchell, The Darkroom Cookbook, 4th Edition, New York, USA, 2016
 D.H.O. John and G.T.J. Field, A Textbook of Photographic Chemistry, Londen, UK: Chapman and Hall Ltd, 1963.
Cover photo is an inverted version of the original “Bonfire Flames” released into the public domain by “He Who Laughs Last”. Original available on Wikimedia Commons.