cooling process, and determined for all kinds of the braneworld ECS. In Section \[SECconcl\], concluding remarks are presented. Throughout the paper, we shall use the high-energy units with $\hbar = c = k_\mathrm{B} = 1$, if not stated otherwise. For simplicity, we assume zero rest energy of neutrinos, isotropic emission at each radius, and the period of evolution of the compact stars, when the temperature is low enough to describe the motion of neutrinos by the null geodesics of the spacetime.

Braneworld neutron stars {#SECbrwns}
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We describe the braneworld neutron stars by the simple model of uniform energy density interior and the “tidally” charged exterior. We give the internal and external geometry and the matching conditions. Accordingly, we determine conditions on the existence of ECS depending on the tidal parameter characterizing the external spacetime.

Matching conditions of internal and external spacetimes
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In the standard Schwarzschild coordinates and the high-energy units, the line element of the spherically symmetric spacetimes reads

$$\mathrm{d}s^2 = -A^{2}(r)\mathrm{d}t^2 + B^{2}(r)\mathrm{d}r^2 + r^{2}\mathrm{d}\Omega.\label{EQmetric}$$

The internal solution is characterized by the uniform energy density distribution — $\varrho = \mathrm{const}$, and by the tension of the brane — $\lambda$. We assume $\varrho > 0$, but we allow 