  Specific heat studies of the Sr$_{1-x}$Ba$_x$Mn$_{1-y}$Ti$_y$O$_3$
polycrystalline samples performed by the relaxation and DSC methods over the
temperature range 2 - 450 K are reported. Anomalies accompanying the
antiferromagnetic-paramagnetic and ferroelectric-paraelectric phase transitions
were measured and analyzed. The system studied is a promising multiferroic
material of the 2$^{\text {nd}}$ type, in which a strong coupling between the
magnetic and electric systems can be related to the fact that the same
Mn$^{4+}$ ions are responsible for the two orderings. Analysis of the anomaly
at the magnetic transition was done by using the advanced theory of the
continuous transitions, in which the presence of higher order terms in the free
energy is considered, and the parameters of the critical behavior of the system
were estimated. It was found that, despite of earlier predictions, the
transition loses the continuous character and can become the 1$^{\text {st}}$
order process for the Ti-containing samples. The anomalies accompanying the
1$^{\text{st}}$ order high-temperature ferroelectric transition were found to
be hardly visible by using both techniques, which was ascribed to a wide
temperature range of coexistence of the dielectric and ferroelectric phases.
