In 1907 he became Boltzmann's successor at the University of Vienna as the head of the Department of Theoretical Physics.  He had a number of illustrious pupils there and had an especially significant impact on Erwin Schrödinger, who later won the Nobel Prize for Physics for his contributions to Quantum Mechanics. 

When the war broke out in 1914, he volunteered at once into the Austria-Hungarian army.  He fought as Oberleutnant against the Italians in Tyrol.  He was wounded, recovered and returned to the front.  He was then killed by a grenade in an attack on Mount Plaut on 7 October 1915 at the age of 40. 

for the so-called "electromagnetic mass", which expresses how much electromagnetic energy contributes to the mass of bodies.  And Henri Poincaré (1900) implicitly used the expression m=E/c2 for the mass of electromagnetic energy. 

Following this line of thought, Hasenöhrl (1904, 1905) published several papers on the inertia of a cavity containing radiation.[H 1][H 2] This was an entirely classical derivation (no use of special relativity) and used Maxwell's equation for the pressure of light.  Hasenöhrl specifically associated the "apparent" mass via inertia with the energy concept through the equation:[H 1]

,

where μ is the apparent mass, E0 is the radiation energy, and the speed of light.  Subsequently, he used the notation:[H 2]

,

where hε0 is the radiation energy.  He also concluded that this result is valid for all radiating bodies, i.e. for all bodies whose temperature is > 0°K.  For this result Hasenöhrl was awarded the Haitinger prize of the Austrian Academy of Sciences.  He wrote in 1904:[H 2]

Since the heat content of every body partly consists of radiating heat, the things that we have demonstrated at a cavity, are true mutatis mutandis for every body whose temperature is different from 0° A.. In particular, every body must have an apparent mass determined by the inner radiation, and which is therefore above all dependent on the temperature. 

However, it was shown by Abraham that Hasenöhrl's calculation for the apparent mass was incorrect, so he published another paper in 1905, where he presented Abraham's criticism and corrected his formula to:[H 3]

In some additional papers (1907, 1908)[H 4] Hasenöhrl elaborated further on his 1904-work and concluded that his new results were now in accordance to the theories of Mosengeil and Planck.  However, he complained about the fact that Planck (1907) did not mention his earlier 1904-results (like the dependency of apparent mass on temperature).  In 1908 Planck wrote that the results of Hasenöhrl's new approach from 1907 were indeed equivalent to those of relativity.[4]

Afterwards, several authors gave credit to Hasenöhrl for his 1904 achievements on cavity radiation. 

Radiation in a moving cavity.  This case is of historic interest, since it can be treated by electrodynamics alone, even without relativity theory.  Then one necessarily comes to ascribe momentum and thus inertial mass to the moving radiation energy.  It's interesting that this result was already found by F. Hasenöhrl before the introduction of relativity theory.  However, his conclusions were in some points in need of correction.  A complete solution of this problem was first given by K. v. Mosengeil.[6]

There are different explanations for this result and its deviation from the relativistic formula .  Enrico Fermi and others argued[7][8] that this problem is analogous to the so-called 4/3 problem of electromagnetic mass.  That is, if Hasenöhrl had included the shell in his calculations in a way consistent with relativity, the pre-factor of 4/3 would have been 1, so yielding .  He could not have done this, since he did not have relativistic mechanics, with which he could model the shell. 

On the other hand, Stephen Boughn and Tony Rothman in 2011[9] (and Boughn in 2012[10]), who gave a historical account of different solutions to the problem, argued that the above explanation is insufficient.  After providing a complete relativistic description and solution of the cavity problem (in the "constant velocity case" and "slow acceleration case"), they wrote:

... more generally the reason he [Hasenöhrl] achieved an incorrect result on both occasions is that he wants to rigorously equate the work performed to kinetic energy, as the work-energy theorem demands.  Unfortunately, he does not know how to properly compute the energy.  In particular, Hasenöhrl does not conceive of the fact that if the radiators are losing energy, they must be losing mass, which contains an element of irony because it is precisely a mass-energy relation that he is trying to establish. [...] 
Let us end by saying that Fritz Hasenöhrl attempted a legitimate thought experiment and tackled it with the tools available at the time.  He was working during a transition period and did not create the new theory necessary to allow him to solve the problem correctly and completely.  Nevertheless, his basic conclusion remained valid and for that he should be given credit. 

published by Albert Einstein in September 1905 in the Annalen der Physik —a few editions after Hasenöhrl published his results on cavity radiation.  The similarity between those formulas led some critics of Einstein, up until the 1930s, to claim that he plagiarized the formula from Hasenöhrl, often in connection with the antisemiticDeutsche Physik. 

As an example, Phillip Lenard published a paper in 1921 in which he gave priority for "E=mc²" to Hasenöhrl (Lenard also gave credit to Johann Georg von Soldner and Paul Gerber in relation to some effects of general relativity).[11] However, Max von Laue quickly rebutted those claims by saying that the inertia of electromagnetic energy was long known before Hasenöhrl, especially by the works of Henri Poincaré (1900) and Max Abraham (1902), while Hasenöhrl only used their results for his calculation on cavity radiation.  Laue continued by saying that credit for establishing the inertia of all forms of energy (the real mass-energy equivalence) goes to Einstein, who was also the first to understand the deep implications of that equivalence in relation to relativity.[12]