  Thermodynamics of the Kitaev honeycomb magnet $\alpha$-RuCl$_3$ is studied
for different directions of in-plane magnetic field using measurements of the
magnetic Gr\"uneisen parameter $\Gamma_B$ and specific heat $C$. We identify
two critical fields $B_c^{\rm AF1}$ and $B_c^{\rm AF2}$ corresponding,
respectively, to a transition between two magnetically ordered states and the
loss of magnetic order toward a quantum paramagnetic state. The $B_c^{AF2}$
phase boundary reveals a narrow region of magnetic fields where inverse melting
of the ordered phase may occur. No additional transitions are detected above
$B_c^{\rm AF2}$ for any direction of the in-plane field, although a shoulder
anomaly in $\Gamma_B$ is observed systematically at $8-10$ T. Large
field-induced entropy effects imply additional low-energy excitations at low
fields and/or strongly field-dependent phonon entropies. Our results establish
universal features of $\alpha$-RuCl$_3$ in high magnetic fields and challenge
the presence of a field-induced Kitaev spin liquid in this material.
