Thermodynamic instability of black holes of third order Lovelock gravity

In this paper, we compute the mass and the temperature of the uncharged black holes of third order Lovelock gravity as well as the entropy using the first law of thermodynamics. We perform a stability analysis by studying the curves of the temperature versus the mass parameter, and find that an intermediate thermodynamically unstable phase exists for black holes with a hyperbolic horizon. This unstable phase for the uncharged topological black holes of third order Lovelock gravity does not exist in lower order Lovelock gravity. We also perform a stability analysis for a spherical, seven-dimensional black hole of Lovelock gravity and find that, while these kinds of black holes for small values of Lovelock coefficients have an intermediate unstable phase, they are stable for large values of Lovelock coefficients. We also find that an intermediate unstable phase exists for these black holes in higher dimensions. This analysis shows that the thermodynamic stability of black holes with curved horizons is not a robust feature of all the generalized theories of gravity.

Figure 1

$f(r)$ vs $r$ for $k=-1$, $n=6$, $a=0.2<a_c$, and $m<m_{\mathrm{ext}}$, $m=m_{\mathrm{ext}}$, and $m>m_{\mathrm{ext}}$ from top to bottom, respectively.

Figure 2

$T$ vs $m$ for $k=-1$, $n=6$, $a=0.2<a_c$.

Figure 3

$f(r)$ vs $r$ for $k=-1$, $n=6$, $a=0.3>a_c$, and $m<m_{1\mathrm{ext}}$, $m=m_{1\mathrm{ext}}$, $m_{1\mathrm{ext}}<m<m_{2\mathrm{ext}}$, $m=m_{2\mathrm{ext}}$, and $m>m_{2\mathrm{ext}}$ from top to bottom, respectively.

Figure 4

$T$ vs $m$ for $k=-1$, $n=6$, $a=0.3>a_c$.

Figure 5

$T$ vs $m$ for $k=-1$, $n=6$, $a=0.4$.

Figure 6

$f(r)$ vs $r$ for $k=-1$, $n=8$.

Figure 7

$T$ vs $r_{+}$ with $k=1$ and $n=6$ for $a<a_c$, $a=a_c=.046$, and $a>a_c$ from top to bottom, respectively.

Figure 8

$T$ vs $r_{+}$ with $k=1$ and $a=0.2$ for $n+1=7$, 8, and 10 from top to bottom, respectively.