The heat-capacity and magnetization measurements under high pressure have been carried out in a ferromagnetic superconductor UGe$_2$. Both measurements were done using a same pressure cell in order to obtain both data for one pressure. Contrary to the heat capacity at ambient pressure, an anomaly is found in the heat capacity at the characteristic temperature $T^{*}$ where the magnetization shows an anomalous enhancement under high pressure where the superconductivity appears. This suggests that a thermodynamic phase transition takes place at $T^{*}$ at least under high pressure slightly below $P_{c}^{*}$ where $T^{*}$ becomes zero. The heat-capacity anomaly associated with the superconducting transition is also investigated, where a clear peak of $C/T$ is observed in a narrow pressure region ($Delta P sim 0.1$ GPa) around $P_{c}^{*}$ contrary to the previous results of the resistivity measurement. Present results suggest the importance of the thermodynamic critical point $P_{c}^{*}$ for the appearance of the superconductivity.
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