  Motivated by an anomaly in $R(D^{(*)})={\rm BR}(\bar{B}\rightarrow D^{(*)}
\tau^-\bar{\nu})/{\rm BR}(\bar{B}\rightarrow D^{(*)} l^-\bar{\nu})$ reported by
BaBar, Belle and LHCb, we study $R(D^{(*)})$ in a general two Higgs doublet
model (2HDM). Although it has been suggested that it is difficult for the 2HDM
to explain the current world average for $R(D^{(*)})$, it would be important to
clarify how large deviations from the standard model predictions for
$R(D^{(*)})$ are possible in the 2HDM. We investigate possible corrections to
$R(D^{(*)})$ in the 2HDM by taking into account various flavor physics
constraints (such as $B_c^-\rightarrow \tau^- \bar{\nu}$, $b\rightarrow
s\gamma$, $b\rightarrow s l^+l^-$, $\Delta m_{B_{d,s}}$, $B_s\rightarrow
\mu^+\mu^-$ and $\tau^+\tau^-$, and $B^-\rightarrow \tau^- \bar{\nu}$), and
find that it would be possible (impossible) to accommodate the 1$\sigma$ region
suggested by the Belle's result when we adopt a constraint ${\rm
BR}(B_c^-\rightarrow \tau^- \bar{\nu})\le30~\%$ (${\rm BR}(B_c^-\rightarrow
\tau^- \bar{\nu})\le10~\%$). We also study productions and decays of heavy
neutral and charged Higgs bosons at the Large Hadron Collider (LHC) experiment
and discuss the constraints and implications at the LHC. We show that in
addition to well-studied production modes $bg\rightarrow tH^-$ and
$gg\rightarrow H/A$, exotic productions of heavy Higgs bosons such as
$cg\rightarrow bH^+,t+H/A$ and $c\bar{b}\rightarrow H^+$ would be significantly
large, and the search for their exotic decay modes such as $H/A\rightarrow
t\bar{c}+c\bar{t},~\mu^\pm\tau^\mp$ and $H^+\rightarrow c\bar{b}$ as well as
$H/A\rightarrow \tau^+\tau^-$ and $H^+\rightarrow \tau^+\nu$ would be important
to probe the interesting parameter regions for $R(D^{(*)})$.
