Crop of the original English translation of the famous April 24, 1865 statement of the first two laws of thermodynamics (laws of the universe) by Rudolf Clausius. [3]
In thermodynamics, the laws of thermodynamics define the rules of temperature equivalence (zeroth law, 1786), energy conservation (first law, 1837), entropy tendencies (second law, 1865), and conditions for an absence of temperature (third law, 1906). [1] The combined law of thermodynamics, sometimes called the Gibbs fundamental equation, is the combination of the four laws in one expression.

In 1952, Norwegian-born American physical chemist Lars Onsager's reciprocal relations (1929) began to be referred to as the fourth law of thermodynamics.

In some marginal publications, especially in economic thermodynamics (e.g. via Nicholas Roegen) and ecological thermodynamics, (e.g. via Corrado Giannantoni) one can also find a fourth law, fifth law, sixth law, or seventh law of thermodynamics, etc., among other variations. The zeroth through the fourth laws of thermodynamics were introduced in chronological order as follows:

0th | 1786
The theoretical framework zeroth law was first enunciated by Scottish physicist Joseph Black in his 1786 Lectures on Chemistry, as such: [6]

“[There exists] a tendency of heat to diffuse itself from any hotter body to the cooler around, until it be distributed among them, in such a manner that none of them are disposed to take any more heat from the rest. The heat is thus brought into a state of equilibrium. This equilibrium is somewhat curious. We find that when all mutual action is ended, a thermometer, applied to any one of the bodies, acquires the same degree of expansion: therefore the temperature of them all is the same, and the equilibrium is universal.”

In this context of the property thermal equilibria, Black typically is cited in prefix histories on the zeroth law. [7] Most references, however, state that Irish physicist James Maxwell’s 1871 law of equal temperatures, stated by Maxwell as “if when two bodies are placed in thermal communication neither of them loses or gains heat, the two bodies are said to have equal temperatures of the same temperature [and] are then said to be in thermal equilibrium”, was the first formulation of what is now called the zeroth law. [4] The actual establishment of this argument to the position of a zeroth law, as “two systems in thermal equilibrium with a third system are in thermal equilibrium with each other”, was first formulated by Ralph Fowler in 1931. [5] It seems, however, that the actual coining of the term “zeroth law” was done jointly by Fowler and Edward Guggenheim in their 1939 Statistical Thermodynamics textbook: [8]

“The concept of temperature. As a natural generalization of experience we introduce the postulate: if to assemblies are each in thermal equilibrium with a third assembly, they are in thermal equilibrium with each other. From this it may be shown to follow that the condition for thermal equilibrium between several assemblies is the equality of a certain single-valued function of the thermodynamic states of the assemblies, which may be called the temperature t, any one of the assemblies being used as a ‘thermometer’ reading the temperature t on a suitable scale. This postulate of the ‘existence of temperature’ could with advantage be known as the zeroth law of thermodynamics.”

 A laws of thermodynamics humor (pgs. 14-15, 16-17) book segment.

1st + 2nd | 1865

The first two laws solidified on April 24th 1865, during a reading at the Philosophical Society of Zurich, when German physicist Rudolf Clausius enunciated the finalized verbal form of his two main principles of the mechanical theory of heat:

(1) the energy of the universe is constant
(2) the entropy of the universe tends to a maximum,

which Clausius declared to be the "fundamental laws of the universe". [3]

3rd | 1906
The third law is associated with the work of German physical chemist Walther Nernst and his 1906 heat theorem. According to students of Nernst, sometime during his lectures in Berlin in the early 1900s, Nernst stated the following, or something to this effect:

“The first law of thermodynamics has been discovered by three scientists: Robert Mayer, James Joule, and Hermann Helmholtz; the second by two scientist: Sadi Carnot and Rudolf Clausius; and, as for the third, well, this I have just done by myself.”

In 1910, German physical chemist Otto Sackur, in a paper submitted to the Annalen, was calling Nernst's heat theorem a “new law of thermodynamics”. This paper was read by Max Planck, who began to assimilate Nernst’s work into his own work on thermodynamics, in particular radiation thermodynamics and quantum mechanics. In the November 1910 preface to the third edition of his Treatise on Thermodynamics, Planck was beginning to indicate that Nernst’s heat theorem was an additional principle of the thermodynamics (to that of the first main principle and the second main principle):

“Should this [heat] theorem, as at present appears likely, be found to hold good in all directions, then thermodynamics will be enriched by a principle whose range, not only from the practical, but also from the theoretical point of view, cannot yet be foreseen.”

In 1923, Nernst’s heat theorem had been enshrined into a law with its incorporation into chapter thirty-one “The Third Law of Thermodynamics” of American physical chemist Gilbert Lewis and Merle Randall’s famous Thermodynamics textbook. [10]

4th | 1920s
In 1952,
Lars Onsager’s 1929 reciprocal relations began to questionably be referred to as the fourth law of thermodynamics. By 1976, several authors had been calling Onsager's relations by the name "fourth law". In the application of thermodynamics to evolution, economics, and ecology, scientists with little fundamental training in thermodynamics have purported other "verbal" fourth laws. These include: Nicholas Georgescu-Roegen (1971), Victor Weisskopf (1977), Howard Odum (1983), Harald Wergeland (1985), Charles Hall (1986), Robert Ayres (1998), Stuart Kauffman (2000), I. M. Kolesnikov (2001), Sven Jorgensen (2002), William Dembski (2002), Ian Stewart (2003), Peter Graham (2003), Eric Schneider (2005), and Jurgen Honig (2007), among others. Others, such as Peter Landsberg (1990), argue that there can be no fourth law and that nobody can formulate a fourth law.

Other
The 2002 nearly-incorrigible work of Corrado Giannantoni makes reference to possible fifth law of thermodynamics and possibly sixth law. There's also Murphy's law of thermodynamics.

Laws of human thermodynamics
See main: Laws of human thermodynamics
A common misconception arrived at when first thinking or theorizing about the conception of a science of "human thermodynamics", is to believe that there are some sort of special "laws of human thermodynamics". The first to arrive at this view was English physicist Charles Galton Darwin in his 1952 book The Next Million Years. [2]

 April 2008 video (2:14) by English physical chemist Peter Atkins on the four laws of thermodynamics.
References
1. Atkins, Peter. (2007). Four Laws - that Drive the Universe. Oxford: Oxford University Press.
2. Darwin, Charles G. (1952). The Next Million Years (pg. 26), (Scribd). London: Rupert Hart-Davis.
3. Clausius, Rudolf. (1865). "On Several Convenient Forms of the Fundamental Equations of the Mechanical Theory of Heat" (ninth memoir), Read at the Philosophical Society of Zurich on the 24th of April, 1865; published in the Vierteljahrsschrift of this society, Bd. x. S. 1.; Pogg. Ann. July, 1865, Bd. cxxv. S. 353; Journ. de Liouville, 2e ser. t. x. p. 361; The Mechanical Theory of Heat, ch. 9 (pg. 365). J. Van Voorst, 1867.
4. (a) Maxwell, James. (1871). Theory of Heat (pgs. 32, 336). London: Longmans, Green and Co.
(b) Pierre, Perrot. (1998). A to Z of Thermodynamics (pg. 328). Oxford University Press.
5. Kestin, Joseph. (1979). A Course in Thermodynamics, Vol. 1. (pg. 40). Taylor and Francis.
6. Black, Joseph. (1786).Lectures on the Elements of Chemistry. University of Edinburgh.
7. Katchalsky, Aharon and Curran, Peter F. (1965). Nonequilibrium Thermodynamics in Biophysics (pg. 6). Harvard University Press.
8. Fowler, Ralph and Guggenheim, Eduard A. (1939). Statistical Thermodynamics: a Version of Statistical Mechanics for Students of Physics and Chemistry (zeroth law: pg. 56 - coined). Cambridge University Press.
9. Nye, Mary Jo (1999). Before Big Science: the Pursuit of Modern Chemistry and Physics, 1800-1940 (ch. 4: Thermodynamics, Thermochemistry, and the Science of Energy, pgs. 88-146). Harvard University Press.
10. Lewis, Gilbert and Randall, Merle. (1923). Thermodynamics and the Free Energy of Chemical Substances (ch. 31: The Third Law of Thermodynamics, pgs. 435-54). McGraw-Hill.