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Motivated by recent developments of black hole thermodynamics in de Rham, Gabadadze, and Tolley (dRGT) massive gravity, we study the critical behaviors of topological Anti-de Sitter (AdS) black holes in the presence of Born-Infeld nonlinear electrodynamics. Here the cosmological constant appears as a dynamical pressure of the system and its corresponding conjugate quantity is interpreted as thermodynamic volume. This shows that, besides the Van der Waals-like SBH/LBH phase transitions, the so-called reentrant phase transition (RPT) appears in four-dimensional space-time when the coupling coefficients

Einstein’s General Relativity (GR), which describes that the graviton is a massless spin-2 particle, helped us to understand the dynamics of the Universe [

In particular, Vainshtein [

Recent development on the thermodynamics of black holes in extended phase space shows that the cosmological constant can be interpreted as the thermodynamic pressure and treated as a thermodynamic variable in its own right [

In Maxwell’s electromagnetic field theory, a point-like charge which allowed a singularity at the charge position usually brings about infinite self-energy. In order to overcome this problem, Born and Infeld [

This paper is organized as follows. In Section

We start with the action of

Consider the metric of

By using the reference metric [

In addition, the electromagnetic field tensor in

Then the mass

In addition, the Hawking temperature which is related to the definition of surface gravity on the outer horizon

The behavior of free energy

For further convenience, we denote

We know that the critical point occurs when

For later discussions, it is convenient to rescale some quantities in the following way:

In what follows we shall specialize to

For

Depending on different values of

When

Now by inserting solutions of

Born-Infeld AdS black holes for

Born-Infeld AdS black holes for

The coexistence line of the VdW-like SBH/LBH phase transition is depicted by a thick solid line. It initiates from the critical point

For

when taking

Born-Infeld AdS black holes for

With regard to

All in all, when the parameter

Then (

When

Here we adopt similar discussions above. The function

Born-Infeld AdS black holes for

In this case, it is hard work to discuss the asymptotic property of (

Born-Infeld AdS black holes for

In [

For the case of

Behavior of

Gibbs free energy of five-dimensional AdS black holes for

Further increasing

In the extended phase space, we have studied the phase transition and critical behavior of topological AdS black holes in the four- and five-dimensional Born-Infeld-massive gravity. For

In some range of the parameters, there are three critical points for five-dimensional Born-Infeld AdS black hole. In such range, the Gibbs free energy displays the behavior of two swallowtails. This phenomenon has been never recovered before.

Recent observations of gravitational waves have put an upper bound of

Reference [

Recently, the charged black hole [

The authors declare that they have no conflicts of interest.

The work is supported by the National Natural Science Foundation of China under Grant nos. 11647050, 11605152, 11675139, and 51575420, Scientific Research Program Funded by Shaanxi Provincial Education Department under Program no. 16JK1394, and Natural Science Foundation of Jiangsu Province under Grant no. BK20160452.